Computational Chemistry: Introduction to the Theory and Applications of Molecular and Quantum Mechanics

Computational Chemistry

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Errol G. Lewars

Computational Chemistry Introduction to the Theory and Applications of Molecular and Quantum Mechanics Second Edition

Prof. Errol G. Lewars Trent University Dept. Chemistry West Bank Drive 1600 K9J 7B8 Peterborough Ontario Canada [email protected]

ISBN 978-90-481-3860-9 e-ISBN 978-90-481-3862-3 DOI 10.1007/978-90-481-3862-3 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2010938715 # Springer ScienceþBusiness Media B.V. 2011 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: KuenkelLopka GmbH Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Anne and John, who know what their contributions were

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Preface

Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry. If mathematical analysis should ever hold a prominent place in chemistry – an aberration which is happily almost impossible – it would occasion a rapid and widespread degeneration of that science. Augustus Compte, French philosopher, 1798–1857; in Philosophie Positive, 1830. A dissenting view: The more progress the physical sciences make, the more they tend to enter the domain of mathematics, which is a kind of center to which they all converge. We may even judge the degree of perfection to which a science has arrived by the facility to which it may be submitted to calculation. Adolphe Quetelet, French astronomer, mathematician, statistician, and sociologist, 1796–1874, writing in 1828. This second edition differs from the first in these ways: 1. The typographical errors that were found in the first edition have been (I hope) corrected. 2. Those equations that should be memorized are marked by an asterisk, for example *(2.1). 3. Sentences and paragraphs have frequently been altered to clarify an explanation. 4. The biographical footnotes have been updated as necessary. 5. Significant developments since 2003, up to near mid-2010, have been added and referenced in the relevant places. 6. Some topics not in first edition, solvation effects, how to do CASSCF calculations, and transition elements, have been added. As might be inferred from the word Introduction, the purpose of this book is to teach the basics of the core concepts and methods of computational chemistry. This is a textbook, and no attempt has been made to please every reviewer by dealing with esoteric “advanced” topics. Some fundamental concepts are the idea of a

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potential energy surface, the mechanical picture of a molecule as used in molecular mechanics, and the Schro¨dinger equation and its elegant taming with matrix methods to give energy levels and molecular orbitals. All the needed matrix algebra is explained before it is used. The fundamental methods of computational chemistry are molecular mechanics, ab initio, semiempirical, and density functional methods. Molecular dynamics and Monte Carlo methods are only mentioned; while these are important, they utilize fundamental concepts and methods treated here. I wrote the book because there seemed to be no text quite right for an introductory course in computational chemistry suitable for a fairly general chemical audience; I hope it will be useful to anyone who wants to learn enough about the subject to start reading the literature and to start doing computational chemistry. There are excellent books on the field, but evidently none that seeks to familiarize the general student of chemistry with computational chemistry in the same sense that standard textbooks of those subjects make organic or physical chemistry accessible. To that end the mathematics has been held on a leash; no attempt is made to prove that molecular orbitals are vectors in Hilbert space, or that a finite-dimensional innerproduct space must have an orthonormal basis, and the only sections that the nonspecialist may justifiably view with some trepidation are the (outlined) derivation of the Hartree–Fock and Kohn–Sham equations. These sections should be read, if only to get the flavor of the procedures, but should not stop anyone from getting on with the rest of the book. Computational chemistry has become a tool used in much the same spirit as infrared or NMR spectroscopy, and to use it sensibly it is no more necessary to be able to write your own programs than the fruitful use of infrared or NMR spectroscopy requires you to be able to able to build your own spectrometer. I have tried to give enough theory to provide a reasonably good idea of how the programs work. In this regard, the concept of constructing and diagonalizing a Fock matrix is introduced early, and there is little talk of secular determinants (except for historical reasons in connection with the simple Hu¨ckel method). Many results of actual computations, most of them specifically for this book, are given. Almost all the assertions in these pages are accompanied by literature references, which should make the text useful to researchers who need to track down methods or results, and students (i.e. anyone who is still learning anything) who wish to delve deeper. The material should be suitable for senior undergraduates, graduate students, and novice researchers in computational chemistry. A knowledge of the shapes of molecules, covalent and ionic bonds, spectroscopy, and some familiarity with thermodynamics at about the level provided by second- or third-year undergraduate courses is assumed. Some readers may wish to review basic concepts from physical and organic chemistry. The reader, then, should be able to acquire the basic theory and a fair idea of the kinds of results to be obtained from the common computational chemistry techniques. You will learn how one can calculate the geometry of a molecule, its IR and UV spectra and its thermodynamic and kinetic stability, and other information needed to make a plausible guess at its chemistry.

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Computational chemistry is accessible. Hardware has become far cheaper than it was even a few years ago, and powerful programs previously available only for expensive workstations have been adapted to run on relatively inexpensive personal computers. The actual use of a program is best explained by its manuals and by books written for a specific program, and the actual directions for setting up the various computations are not given here. Information on various programs is provided in Chapter 9. Read the book, get some programs and go out and do computational chemistry. You may make mistakes, but they are unlikely to put you in the same kind of danger that a mistake in a wet lab might. It is a pleasure acknowledge the help of: Professor Imre Csizmadia of the University of Toronto, who gave unstintingly of his time and experience, The students in my computational and other courses, The generous and knowledgeable people who subscribe to CCL, the computational chemistry list, an exceedingly helpful forum anyone seriously interested in the subject, My editor for the first edition at Kluwer, Dr Emma Roberts, who was always most helpful and encouraging, Professor Roald Hoffmann of Cornell University, for his insight and knowledge on sometimes arcane matters, Professor Joel Liebman of the University of Maryland, Baltimore County for stimulating discussions, Professor Matthew Thompson of Trent University, for stimulating discussions The staff at Springer for the second edition: Dr Sonia Ojo who helped me to initiate the project, and Mrs Claudia Culierat who assumed the task of continuing to assist me in this venture and was always extremely helpful. No doubt some names have been, unjustly, inadvertently omitted, for which I tender my apologies. Ontario, Canada April 2010

E. Lewars

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Contents

1

An Outline of What Computational Chemistry Is All About . . . . . . . . . . . . . 1 1.1 What You Can Do with Computational Chemistry. . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Tools of Computational Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Putting It All Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 The Philosophy of Computational Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Easier Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Harder Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

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The Concept of the Potential Energy Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Stationary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 The Born–Oppenheimer Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Geometry Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Stationary Points and Normal-Mode Vibrations – Zero Point Energy . . . . 30 2.6 Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Easier Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Harder Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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Molecular Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 The Basic Principles of Molecular Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.1 Developing a Forcefield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.2 Parameterizing a Forcefield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2.3 A Calculation Using Our Forcefield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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3.3 Examples of the Use of Molecular Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3.1 To Obtain Reasonable Input Geometries for Lengthier (Ab Initio, Semiempirical or Density Functional) Kinds of Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3.2 To Obtain Good Geometries (and Perhaps Energies) for Small- to Medium-Sized Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3.3 To Calculate the Geometries and Energies of Very Large Molecules, Usually Polymeric Biomolecules (Proteins and Nucleic Acids). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.4 To Generate the Potential Energy Function Under Which Molecules Move, for Molecular Dynamics or Monte Carlo Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.5 As a (Usually Quick) Guide to the Feasibility of, or Likely Outcome of, Reactions in Organic Synthesis . . . . . . . . . . . . . . . . . . . . . . 66 3.4 Geometries Calculated by MM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.5 Frequencies and Vibrational Spectra Calculated by MM . . . . . . . . . . . . . . . 72 3.6 Strengths and Weaknesses of Molecular Mechanics . . . . . . . . . . . . . . . . . . . . 73 3.6.1 Strengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.6.2 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Easier Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Harder Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4

Introduction to Quantum Mechanics in Computational Chemistry . . . . 85 4.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 The Development of Quantum Mechanics. The Schro¨dinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.2.1 The Origins of Quantum Theory: Blackbody Radiation and the Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.2.2 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2.3 Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2.4 The Nuclear Atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2.5 The Bohr Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.2.6 The Wave Mechanical Atom and the Schro¨dinger Equation. . . . . . 96 4.3 The Application of the Schro¨dinger Equation to Chemistry by Hu¨ckel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.3.2 Hybridization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.3.3 Matrices and Determinants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3.4 The Simple Hu¨ckel Method – Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.3.5 The Simple Hu¨ckel Method – Applications . . . . . . . . . . . . . . . . . . . . . 133 4.3.6 Strengths and Weaknesses of the Simple Hu¨ckel Method. . . . . . . 144

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4.3.7 The Determinant Method of Calculating the Hu¨ckel c’s and Energy Levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Extended Hu¨ckel Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 An Illustration of the EHM: the Protonated Helium Molecule. . 4.4.3 The Extended Hu¨ckel Method – Applications . . . . . . . . . . . . . . . . . . . 4.4.4 Strengths and Weaknesses of the Extended Hu¨ckel Method . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Easier Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harder Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Ab initio Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Basic Principles of the Ab initio Method . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 The Hartree SCF Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 The Hartree–Fock Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Basis Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Gaussian Functions; Basis Set Preliminaries; Direct SCF. . . . . . . 5.3.3 Types of Basis Sets and Their Uses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Post-Hartree–Fock Calculations: Electron Correlation . . . . . . . . . . . . . . . . . 5.4.1 Electron Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 The Møller–Plesset Approach to Electron Correlation . . . . . . . . . . 5.4.3 The Configuration Interaction Approach To Electron Correlation – The Coupled Cluster Method . . . . . . . . . . . . . . . . . . . . . 5.5 Applications of the Ab initio Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Frequencies and Vibrational Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Properties Arising from Electron Distribution: Dipole Moments, Charges, Bond Orders, Electrostatic Potentials, Atoms-in-Molecules (AIM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Miscellaneous Properties – UV and NMR Spectra, Ionization Energies, and Electron Affinities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.6 Visualization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Strengths and Weaknesses of Ab initio Calculations . . . . . . . . . . . . . . . . . . . 5.6.1 Strengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Easier Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harder Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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146 152 152 160 163 164 165 168 172 172 175 175 176 176 177 181 232 232 233 238 255 255 261 269 281 281 291 332

337 359 364 372 372 372 373 373 388 389

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Contents

Semiempirical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Basic Principles of SCF Semiempirical Methods. . . . . . . . . . . . . . . . . . 6.2.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 The Pariser-Parr-Pople (PPP) Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 The Complete Neglect of Differential Overlap (CNDO) Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 The Intermediate Neglect of Differential Overlap (INDO) Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 The Neglect of Diatomic Differential Overlap (NDDO) Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Applications of Semiempirical Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Frequencies and Vibrational Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Properties Arising from Electron Distribution: Dipole Moments, Charges, Bond Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Miscellaneous Properties – UV Spectra, Ionization Energies, and Electron Affinities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.6 Visualization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.7 Some General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Strengths and Weaknesses of Semiempirical Methods . . . . . . . . . . . . . . . . . 6.4.1 Strengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Easier Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harder Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density Functional Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Basic Principles of Density Functional Theory . . . . . . . . . . . . . . . . . . . . 7.2.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Forerunners to Current DFT Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Current DFT Methods: The Kohn–Sham Approach . . . . . . . . . . . . . 7.3 Applications of Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Frequencies and Vibrational Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Properties Arising from Electron Distribution – Dipole Moments, Charges, Bond Orders, Atoms-in-Molecules . . . . . . . . . 7.3.5 Miscellaneous Properties – UV and NMR Spectra, Ionization Energies and Electron Affinities, Electronegativity, Hardness, Softness and the Fukui Function. . 7.3.6 Visualization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

391 391 393 393 396 398 399 400 412 412 419 423 426 431 434 435 436 436 436 437 438 443 443 445 445 447 447 448 449 467 468 477 484 487

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7.4 Strengths and Weaknesses of DFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Strengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Easier Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harder Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

9

Some “Special” Topics: Solvation, Singlet Diradicals, A Note on Heavy Atoms and Transition Metals . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Solvation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Ways of Treating Solvation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Singlet Diradicals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Problems with Singlet Diradicals and Model Chemistries . . . . . . 8.2.3 (1) Singlet Diradicals: Beyond Model Chemistries. (2) Complete Active Space Calculations (CAS). . . . . . . . . . . . . . . . . 8.3 A Note on Heavy Atoms and Transition Metals. . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Heavy Atoms and Relativistic Corrections . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Some Heavy Atom Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Transition Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solvation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Easier Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harder Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Singlet Diradicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Easier Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harder Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heavy Atoms and Transition Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Easier Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harder Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selected Literature Highlights, Books, Websites, Software and Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 From the Literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 To the Literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Websites for Computational Chemistry in General. . . . . . . . . . . . . .

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9.3 Software and Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655

Chapter 1

An Outline of What Computational Chemistry Is All About

Knowledge is experiment’s daughter Leonardo da Vinci, in Pensieri, ca. 1492 Nevertheless:

Abstract You can calculate molecular geometries, rates and equilibria, spectra, and other physical properties. The tools of computational chemistry are molecular mechanics, ab initio, semiempirical and density functional methods, and molecular dynamics. Computational chemistry is widely used in the pharmaceutical industry to explore the interactions of potential drugs with biomolecules, for example by docking a candidate drug into the active site of an enzyme. It is also used to investigate the properties of solids (e.g. plastics) in materials science. It does not replace experiment, which remains the final arbiter of truth about Nature.

1.1

What You Can Do with Computational Chemistry

Computational chemistry (also called molecular modelling; the two terms mean about the same thing) is a set of techniques for investigating chemical problems on a computer. Questions commonly investigated computationally are: Molecular geometry: the shapes of molecules – bond lengths, angles and dihedrals. Energies of molecules and transition states: this tells us which isomer is favored at equilibrium, and (from transition state and reactant energies) how fast a reaction should go. Chemical reactivity: for example, knowing where the electrons are concentrated (nucleophilic sites) and where they want to go (electrophilic sites) enables us to predict where various kinds of reagents will attack a molecule. IR, UV and NMR spectra: these can be calculated, and if the molecule is unknown, someone trying to make it knows what to look for. E.G. Lewars, Computational Chemistry, DOI 10.1007/978-90-481-3862-3_1, # Springer ScienceþBusiness Media B.V. 2011

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1 An Outline of What Computational Chemistry Is All About

The interaction of a substrate with an enzyme: seeing how a molecule fits into the active site of an enzyme is one approach to designing better drugs. The physical properties of substances: these depend on the properties of individual molecules and on how the molecules interact in the bulk material. For example, the strength and melting point of a polymer (e.g. a plastic) depend on how well the molecules fit together and on how strong the forces between them are. People who investigate things like this work in the field of materials science.

1.2

The Tools of Computational Chemistry

In studying these questions computational chemists have a selection of methods at their disposal. The main tools available belong to five broad classes: Molecular mechanics is based on a model of a molecule as a collection of balls (atoms) held together by springs (bonds). If we know the normal spring lengths and the angles between them, and how much energy it takes to stretch and bend the springs, we can calculate the energy of a given collection of balls and springs, i.e. of a given molecule; changing the geometry until the lowest energy is found enables us to do a geometry optimization, i.e. to calculate a geometry for the molecule. Molecular mechanics is fast: a fairly large molecule like a steroid (e.g. cholesterol, C27H46O) can be optimized in seconds on a good personal computer. Ab Initio calculations (ab initio, Latin: “from the start”, i.e. from first principles”) are based on the Schr€ odinger equation. This is one of the fundamental equations of modern physics and describes, among other things, how the electrons in a molecule behave. The ab initio method solves the Schr€odinger equation for a molecule and gives us an energy and wavefunction. The wavefunction is a mathematical function that can be used to calculate the electron distribution (and, in theory at least, anything else about the molecule). From the electron distribution we can tell things like how polar the molecule is, and which parts of it are likely to be attacked by nucleophiles or by electrophiles. The Schr€ odinger equation cannot be solved exactly for any molecule with more than one (!) electron. Thus approximations are used; the less serious these are, the “higher” the level of the ab initio calculation is said to be. Regardless of its level, an ab initio calculation is based only on basic physical theory (quantum mechanics) and is in this sense “from first principles”. Ab initio calculations are relatively slow: the geometry and IR spectra (¼ the vibrational frequencies) of propane can be calculated at a reasonably high level in minutes on a personal computer, but a fairly large molecule, like a steroid, could take perhaps days. The latest personal computers, with 2 or more GB of RAM and a thousand or more gigabytes of disk space, are serious computational tools and now compete with UNIX workstations even for the demanding tasks associated with high-level ab initio calculations. Indeed, one now hears little talk of “workstations”, machines costing ca. $15,000 or more [1].

1.3 Putting It All Together

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Semiempirical calculations are, like ab initio, based on the Schr€odinger equation. However, more approximations are made in solving it, and the very complicated integrals that must be calculated in the ab initio method are not actually evaluated in semiempirical calculations: instead, the program draws on a kind of library of integrals that was compiled by finding the best fit of some calculated entity like geometry or energy (heat of formation) to the experimental values. This plugging of experimental values into a mathematical procedure to get the best calculated values is called parameterization (or parametrization). It is the mixing of theory and experiment that makes the method “semiempirical”: it is based on the Schr€odinger equation, but parameterized with experimental values (empirical means experimental). Of course one hopes that semiempirical calculations will give good answers for molecules for which the program has not been parameterized. Semiempirical calculations are slower than molecular mechanics but much faster than ab initio calculations. Semiempirical calculations take roughly 100 times as long as molecular mechanics calculations, and ab initio calculations take roughly 100–1,000 times as long as semiempirical. A semiempirical geometry optimization on a steroid might take seconds on a PC. Density functional calculations (DFT calculations, density functional theory) are, like ab initio and semiempirical calculations, based on the Schr€odinger equation However, unlike the other two methods, DFT does not calculate a conventional wavefunction, but rather derives the electron distribution (electron density function) directly. A functional is a mathematical entity related to a function. Density functional calculations are usually faster than ab initio, but slower than semiempirical. DFT is relatively new (serious DFT computational chemistry goes back to the 1980s, while computational chemistry with the ab initio and semiempirical approaches was being done in the 1960s). Molecular dynamics calculations apply the laws of motion to molecules. Thus one can simulate the motion of an enzyme as it changes shape on binding to a substrate, or the motion of a swarm of water molecules around a molecule of solute; quantum mechanical molecular dynamics also allows actual chemical reactions to be simulated.

1.3

Putting It All Together

Very large biological molecules are studied mainly with molecular mechanics, because other methods (quantum mechanical methods, based on the Schr€odinger equation: semiempirical, ab initio and DFT) would take too long. Novel molecules, with unusual structures, are best investigated with ab initio or possibly DFT calculations, since the parameterization inherent in MM or semiempirical methods makes them unreliable for molecules that are very different from those used in the parameterization. DFT is relatively new and its limitations are still unclear. Calculations on the structure of large molecules like proteins or DNA are done with molecular mechanics. The motions of these large biomolecules can be studied with

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1 An Outline of What Computational Chemistry Is All About

molecular dynamics. Key portions of a large molecule, like the active site of an enzyme, can be studied with semiempirical or even ab initio methods. Moderately large molecules like steroids can be studied with semiempirical calculations, or if one is willing to invest the time, with ab initio calculations. Of course molecular mechanics can be used with these too, but note that this technique does not give information on electron distribution, so chemical questions connected with nucleophilic or electrophilic behaviour, say, cannot be addressed by molecular mechanics alone. The energies of molecules can be calculated by MM, SE, ab initio or DFT. The method chosen depends very much on the particular problem. Reactivity, which depends largely on electron distribution, must usually be studied with a quantummechanical method (SE, ab initio or DFT). Spectra are most reliably calculated by ab initio or DFT methods, but useful results can be obtained with SE methods, and some MM programs will calculate fairly good IR spectra (balls attached to springs vibrate!). Docking a molecule into the active site of an enzyme to see how it fits is an extremely important application of computational chemistry. One could manipulate the substrate with a mouse or a kind of joystick and try to fit it (dock it) into the active site, with a feedback device enabling you to feel the forces acting on the molecule being docked, but automated docking is now standard. This work is usually done with MM, because of the large molecules involved, although selected portions of the biomolecules can be studied by one of the quantum mechanical methods. The results of such docking experiments serve as a guide to designing better drugs, molecules that will interact better with the desired enzymes but be ignored by other enzymes. Computational chemistry is valuable in studying the properties of materials, i.e. in materials science. Semiconductors, superconductors, plastics, ceramics – all these have been investigated with the aid of computational chemistry. Such studies tend to involve a knowledge of solid-state physics and to be somewhat specialized. Computational chemistry is fairly cheap, it is fast compared to experiment, and it is environmentally safe (although the profusion of computers in the last decade has raised concern about the consumption of energy [2] and the disposal of obsolescent machines [3]). It does not replace experiment, which remains the final arbiter of truth about Nature. Furthermore, to make something – new drugs, new materials – one has to go into the lab. However, computation has become so reliable in some respects that, more and more, scientists in general are employing it before embarking on an experimental project, and the day may come when to obtain a grant for some kinds of experimental work you will have to show to what extent you have computationally explored the feasibility of the proposal.

1.4

The Philosophy of Computational Chemistry

Computational chemistry is the culmination (to date) of the view that chemistry is best understood as the manifestation of the behavior of atoms and molecules, and that these are real entities rather than merely convenient intellectual models [4]. It is

References

5

a detailed physical and mathematical affirmation of a trend that hitherto found its boldest expression in the structural formulas of organic chemistry [5], and it is the unequivocal negation of the till recently trendy assertion [6] that science is a kind of game played with “paradigms” [7]. In computational chemistry we take the view that we are simulating the behaviour of real physical entities, albeit with the aid of intellectual models; and that as our models improve they reflect more accurately the behavior of atoms and molecules in the real world.

1.5

Summary

Computational chemistry allows one to calculate molecular geometries, reactivities, spectra, and other properties. It employs: Molecular mechanics – based on a ball-and-springs model of molecules Ab initio methods – based on approximate solutions of the Schr€odinger equation without appeal to fitting to experiment Semiempirical methods – based on approximate solutions of the Schr€odinger equation with appeal to fitting to experiment (i.e. using parameterization) Density functional theory (DFT) methods – based on approximate solutions of the Schr€ odinger equation, bypassing the wavefunction that is a central feature of ab initio and semiempirical methods Molecular dynamics methods study molecules in motion. Ab initio and the faster DFT enable novel molecules of theoretical interest to be studied, provided they are not too big. Semiempirical methods, which are much faster than ab initio or even DFT, can be applied to fairly large molecules (e.g. cholesterol, C27H46O), while molecular mechanics will calculate geometries and energies of very large molecules such as proteins and nucleic acids; however, molecular mechanics does not give information on electronic properties. Computational chemistry is widely used in the pharmaceutical industry to explore the interactions of potential drugs with biomolecules, for example by docking a candidate drug into the active site of an enzyme. It is also used to investigate the properties of solids (e.g. plastics) in materials science.

References 1. Schaefer HF (2001) The cost-effectiveness of PCs. J. Mol. Struct. (Theochem) 573:129–137 2. McKenna P (2006) The waste at the heart of the web. New Scientist 15 December (No. 2582) 3. Environmental Industry News (2008) Old computer equipment can now be disposed in a way that is safe to both human health and the environment thanks to a new initiative launched today at a United Nations meeting on hazardous waste that wrapped up in Bali, Indonesia, 4 Nov 2008

6

1 An Outline of What Computational Chemistry Is All About

4. The physical chemist Wilhelm Ostwald (Nobel Prize 1909) was a disciple of the philosopher Ernst Mach. Like Mach, Ostwald attacked the notion of the reality of atoms and molecules (“Nobel Laureates in Chemistry, 1901–1992”, James LK (ed) American Chemical Society and the Chemical Heritage Foundation, Washington, DC, 1993) and it was only the work of Jean Perrin, published in 1913, that finally convinced him, perhaps the last eminent holdout against the atomic theory, that these entities really existed (Perrin showed that the number of tiny particles suspended in water dropped off with height exactly as predicted in 1905 by Einstein, who had derived an equation assuming the existence of atoms). Ostwald’s philosophical outlook stands in contrast to that of another outstanding physical chemist, Johannes van der Waals, who staunchly defended the atomic/molecular theory and was outraged by the Machian positivism of people like Ostwald. See Ya Kipnis A, Yavelov BF, Powlinson JS (1996) Van der Waals and molecular science. Oxford University Press, New York. For the opposition to and acceptance of atoms in physics see: Lindley D (2001) Boltzmann’s atom. the great debate that launched a revolution in physics. Free Press, New York; and Cercignani C (1998) Ludwig Boltzmann: the man who trusted atoms. Oxford University Press, New York. Of course, to anyone who knew anything about organic chemistry, the existence of atoms was in little doubt by 1910, since that science had by that time achieved significant success in the field of synthesis, and a rational synthesis is predicated on assembling atoms in a definite way 5. For accounts of the history of the development of structural formulas see Nye MJ (1993) From chemical philosophy to theoretical chemistry. University of California Press, Berkeley, CA; Russell CA (1996) Edward Frankland: chemistry, controversy and conspiracy in Victorian England. Cambridge University Press, Cambridge 6. (a) An assertion of the some adherents of the “postmodernist” school of social studies; see Gross P, Levitt N (1994) The academic left and its quarrels with science. John Hopkins University Press, Baltimore, MD; (b) For an account of the exposure of the intellectual vacuity of some members of this school by physicist Alan Sokal’s hoax see Gardner M (1996) Skeptical Inquirer 20(6):14 7. (a) A trendy word popularized by the late Thomas Kuhn in his book – Kuhn TS (1970) The structure of scientific revolutions. University of Chicago Press, Chicago, IL. For a trenchant comment on Kuhn, see ref. [6b]. (b) For a kinder perspective on Kuhn, see Weinberg S (2001) Facing up. Harvard University Press, Cambridge, MA, chapter 17 Added in press: 8. Fantacci S, Amat A, Sgamellotti A (2010) Computational chemistry, art, and our cultural heritage. Acc Chem Res 43:802

Easier Questions 1. What does the term computational chemistry mean? 2. What kinds of questions can computational chemistry answer? 3. Name the main tools available to the computational chemist. Outline (a few sentences for each) the characteristics of each. 4. Generally speaking, which is the fastest computational chemistry method (tool), and which is the slowest? 5. Why is computational chemistry useful in industry? 6. Basically, what does the Schr€ odinger equation describe, from the chemist’s viewpoint? 7. What is the limit to the kind of molecule for which we can get an exact solution to the Schr€ odinger equation?

Harder Questions

7

8. What is parameterization? 9. What advantages does computational chemistry have over “wet chemistry”? 10. Why can’t computational chemistry replace “wet chemistry”?

Harder Questions Discuss the following, and justify your conclusions. 1. Was there computational chemistry before electronic computers were available? 2. Can “conventional” physical chemistry, such as the study of kinetics, thermodynamics, spectroscopy and electrochemistry, be regarded as a kind of computational chemistry? 3. The properties of a molecule that are most frequently calculated are geometry, energy (compared to that of other isomers), and spectra. Why is it more of a challenge to calculate “simple” properties like melting point and density? Hint: is there a difference between a molecule X and the substance X? 4. Is it surprising that the geometry and energy (compared to that of other isomers) of a molecule can often be accurately calculated by a ball-and-springs model (molecular mechanics)? 5. What kinds of properties might you expect molecular mechanics to be unable to calculate? 6. Should calculations from first principles (ab initio) necessarily be preferred to those which make some use of experimental data (semiempirical)? 7. Both experiments and calculations can give wrong answers. Why then should experiment have the last word? 8. Consider the docking of a potential drug molecule X into the active site of an enzyme: a factor influencing how well X will “hold” is clearly the shape of X; can you think of another factor? Hint: molecules consist of nuclei and electrons. 9. In recent years the technique of combinatorial chemistry has been used to quickly synthesize a variety of related compounds, which are then tested for pharmacological activity (S. Borman, Chemical and Engineering News: 2001, 27 August, p. 49; 2000, 15 May, p. 53; 1999, 8 March, p. 33). What are the advantages and disadvantages of this method of finding drug candidates, compared with the “rational design” method of studying, with the aid of computational chemistry, how a molecule interacts with an enzyme? 10. Think up some unusual molecule which might be investigated computationally. What is it that makes your molecule unusual?

Chapter 2

The Concept of the Potential Energy Surface

Everything should be made as simple as possible, but not simpler. Albert Einstein

Abstract The potential energy surface (PES) is a central concept in computational chemistry. A PES is the relationship – mathematical or graphical – between the energy of a molecule (or a collection of molecules) and its geometry. The Born–Oppenheimer approximation says that in a molecule the nuclei are essentially stationary compared to the electrons. This is one of the cornerstones of computational chemistry because it makes the concept of molecular shape (geometry) meaningful, makes possible the concept of a PES, and simplifies the application of the Schr€ odinger equation to molecules by allowing us to focus on the electronic energy and add in the nuclear repulsion energy later; this third point, very important in practical molecular computations, is elaborated on in Chapter 5. Geometry optimization and transition state optimization are explained.

2.1

Perspective

We begin a more detailed look at computational chemistry with the potential energy surface (PES) because this is central to the subject. Many important concepts that might appear to be mathematically challenging can be grasped intuitively with the insight provided by the idea of the PES [1]. Consider a diatomic molecule AB. In some ways a molecule behaves like balls (atoms) held together by springs (chemical bonds); in fact, this simple picture is the basis of the important method molecular mechanics, discussed in Chapter 3. If we take a macroscopic balls-and-spring model of our diatomic molecule in its normal geometry (the equilibrium geometry), grasp the “atoms” and distort the model by stretching or compressing the “bonds”, we increase the potential energy of the molecular model (Fig. 2.1). The stretched or compressed spring possesses energy, by definition, since we moved a force through a distance to distort it. Since the E.G. Lewars, Computational Chemistry, DOI 10.1007/978-90-481-3862-3_2, # Springer ScienceþBusiness Media B.V. 2011

9

10

2 The Concept of the Potential Energy Surface

model is motionless while we hold it at the new geometry, this energy is not kinetic and so is by default potential (“depending on position”). The graph of potential energy against bond length is an example of a potential energy surface. A line is a one-dimensional “surface”; we will soon see an example of a more familiar two-dimensional surface rather than the line of Fig. 2.1. Real molecules behave similarly to, but differ from our macroscopic model in two relevant ways: 1. They vibrate incessantly (as we would expect from Heisenberg’s uncertainty principle: a stationary molecule would have an exactly defined momentum and position) about the equilibrium bond length, so that they always possess kinetic energy (T) and/or potential energy (V): as the bond length passes through the equilibrium length, V ¼ 0, while at the limit of the vibrational amplitude, T ¼ 0; at all other positions both T and V are nonzero. The fact that a molecule is never actually stationary with zero kinetic energy (it always has zero point energy; Section 2.5) is usually shown on potential energy/bond length diagrams by drawing a series of lines above the bottom of the curve (Fig. 2.2) to indicate the possible amounts of vibrational energy the molecule can have (the vibrational levels it can occupy). A molecule never sits at the bottom of the curve, but rather occupies one of the vibrational levels, and in a collection of molecules the levels are populated according to their spacing and the temperature [2]. We will usually ignore the vibrational levels and consider molecules to rest on the actual potential energy curves or (see below) surfaces. 2. Near the equilibrium bond length qe the potential energy/bond length curve for a macroscopic balls-and-spring model or a real molecule is described fairly well by a quadratic equation, that of the simple harmonic oscillator ðE ¼ ð1=2Þ K ðq  qe Þ2 , where k is the force constant of the spring). However, the potential energy deviates from the quadratic (q2) curve as we move away from qe (Fig. 2.2). The deviations from molecular reality represented by this anharmonicity are not important to our discussion. energy

Fig. 2.1 The potential energy surface for a diatomic molecule. The potential energy increases if the bond length q is stretched or compressed away from its equilibrium value qe. The potential energy at qe (zero distortion of the bond length) has been chosen here as the zero of energy

0

qe

bond length, q

2.1 Perspective

11 energy

quadratic curve

. . .

true molecular potential energy curve vibrational levels

0 qe

bond length, q

Fig. 2.2 Actual molecules do not sit still at the bottom of the potential energy curve, but instead occupy vibrational levels. Also, only near qe, the equilibrium bond length, does the quadratic curve approximate the true potential energy curve

Figure 2.1 represents a one-dimensional PES in the two-dimensional graph of E vs. q. A diatomic molecule AB has only one geometric parameter for us to vary, the bond length qAB. Suppose we have a molecule with more than one geometric parameter, for example water: the geometry is defined by two bond lengths and a bond angle. If we reasonably content ourselves with allowing the two bond lengths to be the same, i.e. if we limit ourselves to C2v symmetry (two planes of symmetry and a two-fold symmetry axis; see Section 2.6) then the PES for this triatomic molecule is a graph of E versus two geometric parameters, q1 ¼ the O–H bond length, and q2 ¼ the H–O–H bond angle (Fig. 2.3). Figure 2.3 represents a twodimensional PES (a normal surface is a 2-D object) in the three-dimensional graph; we could make an actual 3-D model of this drawing of a 3-D graph of E versus q1 and q2. We can go beyond water and consider a triatomic molecule of lower symmetry, such as HOF, hypofluorous acid. This has three geometric parameters, the H–O and O–F lengths and the H–O–F angle. To construct a Cartesian PES graph for HOF analogous to that for H2O would require us to plot E vs. q1 ¼ H–O, q2 ¼ O–F, and q3 ¼ angle H–O–F. We would need four mutually perpendicular axes (for E, q1, q2, q3, Fig. 2.4), and since such a four-dimensional graph cannot be constructed in our three-dimensional space we cannot accurately draw it. The HOF PES is a 3-D “surface” of more than two dimensions in 4-D space: it is a hypersurface, and potential energy surfaces are sometimes called potential energy hypersurfaces. Despite the problem of drawing a hypersurface, we can define the equation E ¼ f (q1, q2, q3) as the potential energy surface for HOF, where f is the function that describes how E varies with the q’s, and treat the hypersurface mathematically. For example, in the AB diatomic molecule PES (a line) of Fig. 2.1 the minimum

12

2 The Concept of the Potential Energy Surface energy

O H

H Pmin

.

q2 = H

q2 = 104.5°

O

angle H

q1 = 0.958 Å q1 = O

H bond length

Fig. 2.3 The H2O potential energy surface. The point Pmin corresponds to the minimum-energy geometry for the three atoms, i.e. to the equilibrium geometry of the water molecule Fig. 2.4 To plot energy against three geometric parameters in a Cartesian coordinate system we would need four mutually perpendicular axes. Such a coordinate system cannot be actually constructed in our three-dimensional space. However, we can work with such coordinate systems, and the potential energy surfaces in them, mathematically

energy

q1 q3

q2

potential energy geometry is the point at which dE/dq ¼ 0. On the H2O PES (Fig. 2.3) the minimum energy geometry is defined by the point Pm, corresponding to the equilibrium values of q1 and q2; at this point dE/dq1 ¼ dE/dq2 ¼ 0. Although hypersurfaces cannot be faithfully rendered pictorially, it is very useful to a computational chemist to develop an intuitive understanding of them. This can be gained with the aid of diagrams like Figs. 2.1 and 2.3, where we content ourselves with a line or a two-dimensional surface, in effect using a slice of a multidimensional diagram. This can be understood by analogy: Fig. 2.5 shows how 2-D slices

2.2 Stationary Points

13 O H

energy

H slice parallel to bond length axis

slice parallel to angle axis

2D surface

q2 = H q1 = O

O

angle H

H bond length energy

energy 1D "surface" bond angle

1D "surface" bond length

Fig. 2.5 Slices through a 2D potential energy surface give 1D surfaces. A slice that is parallel to neither axis would give a plot of geometry versus a composite of bond angle and bond length, a kind of average geometry

can be made of the 3-D diagram for water. The slice could be made holding one or the other of the two geometric parameters constant, or it could involve both of them, giving a diagram in which the geometry axis is a composite of more than one geometric parameter. Analogously, we can take a 3-D slice of the hypersurface for HOF (Fig. 2.6) or even a more complex molecule and use an E versus q1, q2 diagram to represent the PES; we could even use a simple 2D diagram, with q representing one, two or all of the geometric parameters. We shall see that these 2D and particularly 3D graphs preserve qualitative and even quantitative features of the mathematically rigorous but unvisualizable E ¼ f(q1, q2, . . . qn) n-dimensional hypersurface.

2.2

Stationary Points

Potential energy surfaces are important because they aid us in visualizing and understanding the relationship between potential energy and molecular geometry, and in understanding how computational chemistry programs locate and characterize structures

14

2 The Concept of the Potential Energy Surface energy

O F

H

Pmin

q2 = O

q1 = O

F bond length

H bond length

Fig. 2.6 A potential energy surface (PES) for HOF. Here the HOF angle is not shown. This picture could represent one of two possibilities: the angle might be the same (some constant, reasonable value) for every calculated point on the surface; this would be an unrelaxed or rigid PES. Alternatively, for each calculated point the geometry might be that for the best angle corresponding to the other two parameters, i.e. the geometry for each calculated point might be fully optimized (Section 2.4); this would be a relaxed PES

of interest. Among the main tasks of computational chemistry are to determine the structure and energy of molecules and of the transition states involved in chemical reactions: our “structures of interest” are molecules and the transition states linking them. Consider the reaction + –

O

O

O transition state

O O isoozone

O O

O ozone

O

reaction (2.1)

A priori, it seems reasonable that ozone might have an isomer (call it isoozone) and that the two could interconvert by a transition state as shown in Reaction (2.1). We can depict this process on a PES. The potential energy E must be plotted against only two geometric parameters, the bond length (we may reasonably assume that the two O–O bonds of ozone are equivalent, and that these bond lengths remain equal throughout the reaction) and the O–O–O bond angle. Figure 2.7 shows the PES for Reaction (2.1), as calculated by the AM1 semiempirical method (Chapter 6; the AM1 method is unsuitable for quantitative treatment of this problem, but the potential energy surface shown makes the point), and shows how a 2D slice from

2.2 Stationary Points

15 transition state

2.0

0-0

,Å

–1

ol

KJ m

160 140 16

120

1.3

100

0

80

O-O

6 1.1

-O a ngle

1.0

60

, de

gre

es

120.9°

60° relative minimum iSOOZONE

global minimum OZONE O

energy O

O O O

O O

O

O

intrinsic reaction coordinate (IRC)

Fig. 2.7 The ozone/isoozone potential energy surface (calculated by the AM1 method; Chapter 6), a 2D surface in a 3D diagram. The dashed line on the surface is the reaction coordinate (intrinsic reaction coordinate, IRC). A slice through the reaction coordinate gives a 1D “surface” in a 2D diagram. The diagram is not meant to be quantitatively accurate

this 3D diagram gives the energy/reaction coordinate type of diagram commonly used by chemists. The slice goes along the lowest-energy path connecting ozone, isoozone and the transition state, that is, along the reaction coordinate, and the horizontal axis (the reaction coordinate) of the 2D diagram is a composite of O–O bond length and O–O–O angle. In most discussions this horizontal axis is left quantitatively undefined; qualitatively, the reaction coordinate represents the

16

2 The Concept of the Potential Energy Surface

progress of the reaction. The three species of interest, ozone, isoozone, and the transition state linking these two, are called stationary points. A stationary point on a PES is a point at which the surface is flat, i.e. parallel to the horizontal line corresponding to the one geometric parameter (or to the plane corresponding to two geometric parameters, or to the hyperplane corresponding to more than two geometric parameters). A marble placed on a stationary point will remain balanced, i.e. stationary (in principle; for a transition state the balancing would have to be exquisite indeed). At any other point on a potential surface the marble will roll toward a region of lower potential energy. Mathematically, a stationary point is one at which the first derivative of the potential energy with respect to each geometric parameter is zero1: @E @E ¼ ¼  ¼ 0 @ q1 @ q2

(*2.1)

Partial derivatives, ∂E/∂q, are written here rather than dE/dq, to emphasize that each derivative is with respect to just one of the variables q of which E is a function. Stationary points that correspond to actual molecules with a finite lifetime (in contrast to transition states, which exist only for an instant), like ozone or isoozone, are minima, or energy minima: each occupies the lowest-energy point in its region of the PES, and any small change in the geometry increases the energy, as indicated in Fig. 2.7. Ozone is a global minimum, since it is the lowest-energy minimum on the whole PES, while isoozone is a relative minimum, a minimum compared only to nearby points on the surface. The lowest-energy pathway linking the two minima, the reaction coordinate or intrinsic reaction coordinate (IRC; dashed line in Fig. 2.7) is the path that would be followed by a molecule in going from one minimum to another should it acquire just enough energy to overcome the activation barrier, pass through the transition state, and reach the other minimum. Not all reacting molecules follow the IRC exactly: a molecule with sufficient energy can stray outside the IRC to some extent [3]. Inspection of Fig. 2.7 shows that the transition state linking the two minima represents a maximum along the direction of the IRC, but along all other directions it is a minimum. This is a characteristic of a saddle-shaped surface, and the transition state is called a saddle point (Fig. 2.8). The saddle point lies at the “center” of the saddle-shaped region and is, like a minimum, a stationary point, since the PES at that point is parallel to the plane defined by the geometry parameter axes: we can see that a marble placed (precisely) there will balance. Mathematically, minima and saddle points differ in that although both are stationary points (they have zero first derivatives; Eq. 2.1), a minimum is a minimum in all directions, but a saddle point is a maximum along the reaction coordinate and a minimum in all other directions (examine Fig. 2.8). Recalling that minima and maxima can be distinguished by their second derivatives, we can write:

1

Equations marked with an asterisk are those which should be memorized.

2.2 Stationary Points

17

energy transition state transition state region

reaction coordinate minimum

Fig. 2.8 A transition state or saddle point and a minimum. At both the transition state and the minimum ∂E/∂q ¼ 0 for all geometric coordinates q (along all directions). At the transition state ∂E2/∂q2 < 0 for q ¼ the reaction coordinate and > 0 for all other q (along all other directions). At a minimum ∂E2/∂q2 > 0 for all q (along all directions)

For a minimum @2E >0 @q2

(*2.2)

@2E >0 @q2

(*2.3)

for all q. For a transition state

for all q, except along the reaction coordinate, and @2E 0 for all q) and transition states or first-order saddle points; ∂2E/∂qiqj < 0 for one q, along the reaction coordinate (intrinsic reaction coordinate, IRC), and > 0 for all other q. Chemistry is the study of PES stationary points and the pathways connecting them. The Born–Oppenheimer approximation says that in a molecule the nuclei are essentially stationary compared to the electrons. This is one of the cornerstones of computational chemistry because it makes the concept of molecular shape

40

2 The Concept of the Potential Energy Surface

(geometry) meaningful, makes possible the concept of a PES, and simplifies the application of the Schr€ odinger equation to molecules by allowing us to focus on the electronic energy and add in the nuclear repulsion energy later; this third point, very important in practical molecular computations, is elaborated on in Chapter 5. Geometry optimization is the process of starting with an input structure “guess” and finding a stationary point on the PES. The stationary point found will normally be the one closest to the input structure, not necessarily the global minimum. A transition state optimization usually requires a special algorithm, since it is more demanding than that required to find a minimum. Modern optimization algorithms use analytic first derivatives and (usually numerical) second derivatives. It is usually wise to check that a stationary point is the desired species (a minimum or a transition state) by calculating its vibrational spectrum (its normal-mode vibrations). The algorithm for this works by calculating an accurate Hessian (force constant matrix) and diagonalizing it to give a matrix with the “direction vectors” of the normal modes, and a diagonal matrix with the force constants of these modes. A procedure of “mass-weighting” the force constants gives the normal-mode vibrational frequencies. For a minimum all the vibrations are real, while a transition state has one imaginary vibration, corresponding to motion along the reaction coordinate. The criteria for a transition state are appearance, the presence of one imaginary frequency corresponding to the reaction coordinate, and an energy above that of the reactant and the product. Besides serving to characterize the stationary point, calculation of the vibrational frequencies enables one to predict an IR spectrum and provides the zero-point energy (ZPE). The ZPE is needed for accurate comparisons of the energies of isomeric species. The accurate Hessian required for calculation of frequencies and ZPE’s can be obtained either numerically or analytically (faster, but much more demanding of hard drive space).

References 1. (a) Shaik SS, Schlegel HB, Wolfe S (1992) Theoretical aspects of physical organic chemistry: the SN2 mechanism. Wiley, New York. See particularly Introduction and chapters 1 and 2. (b) Marcus RA (1992) Science 256:1523. (c) For a very abstract and mathematical but interesting treatment, see Mezey PG (1987) Potential energy hypersurfaces. Elsevier, New York. (d) Steinfeld JI, Francisco JS, Hase WL (1999) Chemical kinetics and dynamics, 2nd edn. Prentice Hall, Upper Saddle River, NJ 2. Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ, section 4.3 3. Shaik SS, Schlegel HB, Wolfe S (1992) Theoretical aspects of physical organic chemistry: the SN2 mechanism. Wiley, New York, pp 50–51 4. Houk KN, Li Y, Evanseck JD (1992) Angew Chem Int Ed Engl 31:682 5. Atkins P (1998) Physical chemistry, 6th edn. Freeman, New York, pp 830–844 6. Marcelin R (1915) Ann Phys 3:152. Potential energy surface, p 158 7. Eyring H (1935) J Chem Phys 3:107 8. Eyring H, Polanyi M (1931) Z Phys Chem B12:279

References

41

9. (a) Carpenter BK (1992) Acc Chem Res 25:520. (b) Carpenter BK (1997) Am Sci March– April:138. (c) Carpenter BK (1998) Angew Chem Int Ed 37:3341. (d) Reyes MB, Carpenter BK (2000) J Am Chem Soc 122:10163. (e) Reyes MB, Lobkovsky EB, Carpenter BK (2002) J Am Chem Soc 124:641. (f) Nummela J, Carpenter BK (2002) J Am Chem Soc 124:8512. (g) Carpenter BK (2003) J Phys Org Chem 16:858. (h) Litovitz AE, Keresztes I, Carpenter BK (2008) J Am Chem Soc 130:12085 10. Born M, Oppenheimer JR (1927) Ann Phys 84:457 11. A standard molecular surface, corresponding to the size as determined experimentally (e.g. by X-ray diffraction) encloses about 98 per cent of the electron density. See e.g. Bader RFW, Carroll MT, Cheeseman MT, Chang, C (1987) J Am Chem Soc 109:7968 12. (a) For some rarefied but interesting ideas about molecular shape see Mezey PG (1993) Shape in chemistry. VCH, New York. (b) An antimatter molecule lacking definite shape: Surko CM (2007) Nature 449:153. (c) The Cl + H2 reaction: Foreword: Bowman JL (2008) Science 319:40; Garand E, Zhou J, Manolopoulos DE, Alexander MH, Neumark DM (2008) Science 319:72; Erratum 320:612. (d) Baer M (2006) Beyond Born–Haber. Wiley, Hoboken, NJ 13. Zhang XK, Parnis JM, Lewars EG, March RE (1997) Can J Chem 75:276 14. See e.g. Cramer C (2004) Essentials of computational chemistry, 2nd edn. Wiley, Chichester, UK, Section 2.4.1 15. Hehre WJ (1995) Practical strategies for electronic structure calculations. Wavefunction Inc., Irvine, CA, p 9 16. Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ, p 65 17. Scott AP, Radom L (1996) J Phys Chem 100:16502 18. Foresman JB, Frisch Æ (1996) Exploring chemistry with electronic structure methods, 2nd edn. Gaussian Inc., Pittsburgh, PA, pp 173–211 19. Atkins P (1998) Physical chemistry, 6th edn. Freeman, New York, chapter 15 20. Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ, chapter 12 Added in press: 21. Kraka E, Cremer D (2010) Review of computational approaches to the potential energy surface and some new twists, the unified reaction valley approach URVA. Acc Chem Res 43:591–601

42

2 The Concept of the Potential Energy Surface

Easier Questions 1. What is a potential energy surface (give the two viewpoints)? 2. Explain the difference between a relaxed PES and a rigid PES. 3. What is a stationary point? What kinds of stationary points are of interest to chemists, and how do they differ? 4. What is a reaction coordinate? 5. Show with a sketch why it is not correct to say that a transition state is a maximum on a PES. 6. What is the Born–Oppenheimer approximation, and why is it important? 7. Explain, for a reaction A ! B, how the potential energy change on a PES is related to the enthalpy change of the reaction. What would be the problem with calculating a free energy/geometry surface? Hint: Vibrational frequencies are normally calculated only for stationary points. 8. What is geometry optimization? Why is this process for transition states (often called transition state optimization) more challenging than for minima? 9. What is a Hessian? What uses does it have in computational chemistry? 10. Why is it usually good practice to calculate vibrational frequencies where practical, although this often takes considerably longer than geometry optimization?

Harder Questions 1. The Born–Oppenheimer principle is often said to be a prerequisite for the concept of a potential energy surface. Yet the idea of a potential energy surface (Marcelin 1915) predates the Born–Oppenheimer principle (1927). Discuss. 2. How high would you have to lift a mole of water for its gravitational potential energy to be equivalent to the energy needed to dissociate it completely into hydroxyl radicals and hydrogen atoms? The strength of the O–H bond is about 400 kJ mol1; the gravitational acceleration g at the Earth’s surface (and out to hundreds of kilometres) is about 10 m s2. What does this indicate about the role of gravity in chemistry? 3. If gravity plays no role in chemistry, why are vibrational frequencies different for, say, C–H and C–D bonds? 4. We assumed that the two bond lengths of water are equal. Must an acyclic molecule AB2 have equal A–B bond lengths? What about a cyclic molecule AB2? 5. Why are chemists but rarely interested in finding and characterizing secondorder and higher saddle points (hilltops)? 6. What kind(s) of stationary points do you think a second-order saddle point connects?

Harder Questions

43

7. If a species has one calculated frequency very close to 0 cm1 what does that tell you about the (calculated) potential energy surface in that region? 8. The ZPE of many molecules is greater than the energy needed to break a bond; for example, the ZPE of hexane is about 530 kJ mol1, while the strength of a C–C or a C–H bond is only about 400 kJ mol1. Why then do such molecules not spontaneously decompose? 9. Only certain parts of a potential energy surface are chemically interesting: some regions are flat and featureless, while yet other parts rise steeply and are thus energetically inaccessible. Explain. 10. Consider two potential energy surfaces for the HCN ⇌ HNC reaction: A, a plot of energy versus the H–C bond length, and B, a plot of energy versus the HNC angle. Recalling that HNC is the higher-energy species, sketch qualitatively the diagrams for A and B.

Chapter 3

Molecular Mechanics

We don’t give a damn where the electrons are. Words to the author, from the president of a well-known chemical company, emphasizing his firm’s position on basic research

Abstract Molecular mechanics (MM) rests on a view of molecules as balls held together by springs. The potential energy of a molecule can be written as the sum of terms involving bond stretching, angle bending, dihedral angles and nonbonded interactions. Giving these terms explicit mathematical forms constitutes devising a forcefield, and giving actual numbers to the constants in the forcefield constitutes parameterizing the field. An example is given of the devising and parameterization of an MM forcefield. Calculations on biomolecules is a very important application of MM, and the pharmaceutical industry designs new drugs with the aid of MM. Organic synthesis now makes considerable use of MM, which enables chemists to estimate which products are likely to be favored and to devise more realistic routes to a target molecule. In molecular dynamics MM is used to generate the forces acting on molecules and hence to calculate their motions.

3.1

Perspective

Molecular mechanics (MM) [1] is based on a mathematical model of a molecule as a collection of balls (corresponding to the atoms) held together by springs (corresponding to the bonds) (Fig. 3.1). Within the framework of this model, the energy of the molecule changes with geometry because the springs resist being stretched or bent away from some “natural” length or angle, and the balls resist being pushed too closely together. The mathematical model is thus conceptually very close to the intuitive feel for molecular energetics that one obtains when manipulating molecular models of plastic or metal: the model resists distortions (it may break!) from the “natural” geometry that corresponds to the bond lengths E.G. Lewars, Computational Chemistry, DOI 10.1007/978-90-481-3862-3_3, # Springer ScienceþBusiness Media B.V. 2011

45

46

3 Molecular Mechanics

H

H H

C

C C

H H

H H

H

Fig. 3.1 Molecular mechanics (the forcefield method) considers a molecule to be a collection of balls (the atoms) held together by springs (the bonds)

and angles imposed by the manufacturer, and in the case of space-filling models atoms cannot be forced too closely together. The MM model clearly ignores electrons. The principle behind MM is to express the energy of a molecule as a function of its resistance toward bond stretching, bond bending, and atom crowding, and to use this energy equation to find the bond lengths, angles, and dihedrals corresponding to the minimum-energy geometry – or more precisely, to the various possible potential energy surface minima (Chapter 2). In other words, MM uses a conceptually mechanical model of a molecule to find its minimum-energy geometry (for flexible molecules, the geometries of the various conformers). The form of the mathematical expression for the energy, and the parameters in it, constitute a forcefield, and molecular mechanics methods are sometimes called forcefield methods. The term arises because the negative of the first derivative of the potential energy of a particle with respect to displacement along some direction is the force on the particle; a “forcefield” E(x, y, z coordinates of atoms) can be differentiated to give the force on each atom. The method makes no reference to electrons, and so cannot (except by some kind of empirical algorithm) throw light on electronic properties like charge distributions or nucleophilic and electrophilic behaviour. Note that MM implicitly uses the Born–Oppenheimer approximation, for only if the nuclei experience what amounts to a static attractive force, whether from electrons or springs, does a molecule have a distinct geometry (Section 2.3). An important point, which students sometimes have a problem with, is that the concept of a bond is central to MM, but not essential – although often useful – in electronic structure calculations. In MM a molecule is defined by the atoms and the bonds, which latter are regarded almost literally as springs holding the atoms together. Usually, bonds are placed where the rules for drawing structural formulas require them, and to do a MM calculation you must specify each bond as single, double, etc., since this tells the program how strong a bond to use (Sections 3.2.1 and 3.2.2). In an electronic structure calculation–ab initio (Chapter 5), semiempirical (Chapter 6), and density functional theory (Chapter 7) – a molecule is defined by the relative positions of its atomic nuclei, the charge, and the “multiplicity” (which

3.1 Perspective

47

follows easily from the number of unpaired electrons). An oxygen nucleus and two protons with the right x, y, z coordinates, enough electrons for no charge, and multiplicity one (no unpaired electrons) is a water molecule. There is no need to mention bonds here, although the chemist might wish to somehow extract this useful concept from this picture of nuclei and electrons. This can be done by calculating the electron density and associating a bond with, for example, a path along which electron density is concentrated, but there is no unique definition of a bond in electronic structure theory. It is worth noting, too, that in some graphical interfaces used in computational chemistry bonds are specified by the user, while in others they are shown by the program depending on the separation of pairs of atoms. The novice may find it disconcerting to see a specified bond still displayed even when a change in geometry has moved a pair of atoms far apart, or to see a bond vanish when a pair has moved beyond the distance recognized by some fudge factor. Historically [2], molecular mechanics seems to have begun as an attempt to obtain quantitative information about chemical reactions at a time when the possibility of doing quantitative quantum mechanical (Chapter 4) calculations on anything much bigger than the hydrogen molecule seemed remote. Specifically, the principles of MM, as a potentially general method for studying the variation of the energy of molecular systems with their geometry, were formulated in 1946 by Westheimer1 and Meyer [3a], and by Hill [3b]. In this same year Dostrovsky, Hughes2 and Ingold3 independently applied molecular mechanics concepts to the quantitative analysis of the SN2 reaction, but they do not seem to have recognized the potentially wide applicability of this approach [3c]. In 1947 Westheimer [3d] published detailed calculations in which MM was used to estimate the activation energy for the racemization of biphenyls. Major contributors to the development of MM have been Schleyer4 [2b, c] and Allinger5 [1c, d]; one of Allinger’s publications on MM [1d] is, according to the Citation Index, one of the most frequently cited chemistry papers. The Allinger group has, since the 1960s, been responsible for the development of the “MM-series” of programs, commencing with MM1 and continuing with the currently widely-used MM2 and MM3, and MM4 [4]. MM programs [5] like Sybyl and UFF will handle molecules involving much of the periodic table, albeit with some loss of accuracy that one might expect for trading breadth for depth, and MM is

1

Frank H. Westheimer, born Baltimore, Maryland, 1912. Ph.D. Harvard 1935. Professor University of Chicago, Harvard. Died 2007. 2 Edward D. Hughes, born Wales, 1906. Ph.D. University of Wales, D.Sc. University of London. Professor, London. Died 1963. 3 Christopher K. Ingold, born London 1893. D.Sc. London 1921. Professor Leeds, London. Knighted 1958. Died London 1970. 4 Paul von R. Schleyer, born Cleveland, Ohio, 1930. Ph.D. Harvard 1957. Professor Princeton; institute codirector and professor University of Erlangen-N€ urnberg, 1976–1998. Professor University of Georgia. 5 Norman L. Allinger, born Rochester New York, 1930. Ph.D. University of California at Los Angeles, 1954. Professor Wayne State University, University of Georgia.

48

3 Molecular Mechanics

the most widely-used method for computing the geometries and energies of large biological molecules like proteins and nucleic acids (although recently semiempirical (Chapter 6) and even ab initio (Chapter 5) methods have begun to be applied to these large molecules.

3.2 3.2.1

The Basic Principles of Molecular Mechanics Developing a Forcefield

The potential energy of a molecule can be written E¼

X

Estretch þ

bonds

X

X

Ebend þ

angles

dihedrals

Etorsion þ

X

Enonbond

(*3.1)

pairs

where Estretch etc. are energy contributions from bond stretching, angle bending, torsional motion (rotation) around single bonds, and interactions between atoms or groups which are nonbonded (not directly bonded together). The sums are over all the bonds, all the angles defined by three atoms A–B–C, all the dihedral angles defined by four atoms A–B–C–D, and all pairs of significant nonbonded interactions. The mathematical form of these terms and the parameters in them constitute a particular forcefield. We can make this clear by being more specific; let us consider each of these four terms. leq aeq l

a Δ l = l – l eq

+ Δ a = a – a eq

energy

0

Δ l or Δ a

Fig. 3.2 Changes in bond lengths or in bond angles result in changes in the energy of a molecule. Such changes are handled by the Estretch and Ebend terms in the molecular mechanics forcefield. The energy is approximately a quadratic function of the change in bond length or angle

3.2 The Basic Principles of Molecular Mechanics

49

The Bond Stretching Term The increase in the energy of a spring (remember that we are modelling the molecule as a collection of balls held together by springs) when it is stretched (Fig. 3.2) is approximately proportional to the square of the extension: DEstretch ¼ kstretch ðl  leq Þ2

kstretch ¼ the proportionality constant (actually one-half the force constant of the spring or bond [6]; but note the warning about identifying MM force constants with the traditional force constant from, say, spectroscopy – see Section 3.3); the bigger kstretch, the stiffer the bond/spring – the more it resists being stretched. l ¼ length of the bond when stretched. leq ¼ equilibrium length of the bond, its “natural” length. If we take the energy corresponding to the equilibrium length leq as the zero of energy, we can replace DEstretch by Estretch: Estretch ¼ kstretch ðl  leq Þ2

(*3.2)

The Angle Bending Term The increase in energy of system ball-spring-ballspring-ball, corresponding to the triatomic unit A–B–C (the increase in “angle energy”) is approximately proportional to the square of the increase in the angle (Fig. 3.2); analogously to Eq. 3.2: Ebend ¼ kbend ða  aeq Þ2

(*3.3)

kbend ¼ a proportionality constant (one-half the angle bending force constant [6]; note the warning about identifying MM force constants with the traditional force constant from, say, spectroscopy – see Section 3.3)) a ¼ size of the angle when distorted aeq ¼ equilibrium size of the angle, its “natural” value. The Torsional Term Consider four atoms sequentially bonded: A–B–C–D (Fig. 3.3). The dihedral angle or torsional angle of the system is the angle between the A–B bond and the C–D bond as viewed along the B–C bond. Conventionally this angle is considered positive if regarded as arising from clockwise rotation (starting with A–B covering or eclipsing C–D) of the back bond (C–D) with respect to the front bond (A–B). Thus in Fig. 3.3 the dihedral angle A–B–C–D is 60 (it could also be considered as being 300 ). Since the geometry repeats itself every 360 , the energy varies with the dihedral angle in a sine or cosine pattern, as shown in Fig. 3.4 for the simple case of ethane. For systems A–B–C–D of lower symmetry, like butane (Fig. 3.5), the torsional potential energy curve is more complicated, but a combination of sine or cosine functions will reproduce the curve: Etorsion ¼ ko þ

n X r¼1

kr ½1 þ cosðryÞ

(*3.4)

50

3 Molecular Mechanics

A

D B

rotate C-D bond about the B-C bond

A

D B

C

AD

C

A 60° D

B

C

B

dihedral angle = 0°

C

dihedral angle = 60°

Fig. 3.3 Dihedral angles (torsional angles) affect molecular geometries and energies. The energy is a periodic (cosine or combination of cosine functions) function of the dihedral angle; see e.g. Figs. 3.4 and 3.5 H energy kJ mol–1

12 kJ mol–1

H C

H

C H H

H

10

0

D3h

60

D3d

120

180

D3h

D3d

HCCH dihedral, degrees

Fig. 3.4 Variation of the energy of ethane with dihedral angle. The curve can be represented as a cosine function

The Nonbonded Interactions Term This represents the change in potential energy with distance apart of atoms A and B that are not directly bonded (as in A–B) and are not bonded to a common atom (as in A–X–B); these atoms, separated by at least two atoms (A–X–Y–B) or even in different molecules, are said to be nonbonded (with respect to each other). Note that the A-B case is accounted for by the bond stretching term Estretch, and the A–X–B term by the angle bending term Ebend, but the nonbonded term Enonbond is, for the A–X–Y–B case, superimposed upon the torsional term Etorsion: we can think of Etorsion as representing some factor inherent to resistance to rotation about a (usually single) bond X–Y (MM does not attempt to explain the theoretical, electronic basis of this or any other effect), while for certain atoms attached to X and Y there may also be nonbonded interactions.

3.2 The Basic Principles of Molecular Mechanics energy kJ mol–1

51

25 kJ mol–1

20 14 kJ mol–1 10

3 kJ mol–1 0 Me Me

60

120

Me

180 Me

Me

Me

Me H3C

CCCC dihedral, degrees

Me CH3

C

H H

C

H H

Fig. 3.5 Variation of the energy of butane with dihedral angle. The curve can be represented by a sum of cosine functions

The potential energy curve for two nonpolar nonbonded atoms has the general form shown in Fig. 3.6. A simple way to approximate this is by the so-called Lennard-Jones 12–6 potential [7]: Enonbond ¼ knb

   s 12 s6  r r

(*3.5)

r ¼ the distance between the centers of the nonbonded atoms or groups. The function reproduces the small attractive dip in the curve (represented by the negative term) as the atoms or groups approach one another, then the very steep rise in potential energy (represented by the positive, repulsive term raised to a large power) as they are pushed together closer than their van der Waals radii. Setting dE/ dr ¼ 0, we find that for the energy minimum in the curve the corresponding value of r is rmin ¼ 21=6 s, i.e:

s ¼ 21=6 rmin

(3.6)

If we assume that this minimum corresponds to van der Waals contact of the nonbonded groups, then rmin ¼ (RA + RB), the sum of the van der Waals radii of the groups A and B. So 21=6 s ¼ ðRA þ RB Þ

52

3 Molecular Mechanics

Fig. 3.6 Variation of the energy of a molecule with separation of nonbonded atoms or groups. Atoms/ groups A and B may be in the same molecule (as indicated here) or the interaction may be intermolecular. The minimum energy occurs at van der Waals contact. For small nonpolar atoms or groups the minimum energy point represents a drop of a few kJ mol1 (Emin ¼ 1.2 kJ mol1 for CH4/CH4), but short distances can make nonbonded interactions destabilize a molecule by many kJ mol1

energy

0

r Emin rmin = (RA + RB)

r

A

.

RA

.

RB

B

and so s ¼ 21=6 ðRA þ RB Þ ¼ 0:89 ðRA þ RB Þ

(3.7)

Thus s can be calculated from rmin or estimated from the van der Waals radii. Setting E ¼ 0, we find that for this point on the curve r ¼ s, i.e: s ¼ rðE ¼ 0Þ

(3.8)

If we set r ¼ rmin ¼ 21=6 s (from Eq. 3.6) in Eq. 3.5, we find Eðr ¼ rmin Þ ¼ ð1=4Þknb i.e. knb ¼ 4Eðr ¼ rmin Þ

(3.9)

So knb can be calculated from the depth of the energy minimum. In deciding to use equations of the form (3.2), (3.3), (3.4) (3.5) we have decided on a particular MM forcefield. There are many alternative forcefields. For example,

3.2 The Basic Principles of Molecular Mechanics

53

we might have chosen to approximate Estretch by the sum of a quadratic and a cubic term: Estretch ¼ kstretch ðl  leq Þ2 þ kðl  leq Þ3 This gives a somewhat more accurate representation of the variation of energy with length. Again, we might have represented the nonbonded interaction energy by a more complicated expression than the simple 12–6 potential of Eq. 3.5 (which is by no means the best form for nonbonded repulsions). Such changes would represent changes in the forcefield.

3.2.2

Parameterizing a Forcefield

We can now consider putting actual numbers, kstretch, leq, kbend, etc., into Eqs. 3.2, 3.3, 3.4 and 3.5, to give expressions that we can actually use. The process of finding these numbers is called parameterizing (or parametrizing) the forcefield. The set of molecules used for parameterization, perhaps 100 for a good forcefield, is called the training set. In the purely illustrative example below we use just ethane, methane and butane. Parameterizing the Bond Stretching Term A forcefield can be parameterized by reference to experiment (empirical parameterization) or by getting the numbers from high-level ab initio or density functional calculations, or by a combination of both approaches. For the bond stretching term of Eq. 3.2 we need kstretch and leq. Experimentally, kstretch could be obtained from IR spectra, as the stretching frequency of a bond depends on the force constant (and the masses of the atoms involved) [8], and leq could be derived from X-ray diffraction, electron diffraction, or microwave spectroscopy [9]. Let us find kstretch for the C/C bond of ethane by ab initio (Chapter 5) calculations. Normally high-level ab initio calculations would be used to parameterize a forcefield, but for illustrative purposes we can use the low-level but fast STO-3G method [10]. Equation 3.2 shows that a plot of Estretch against (l–leq)2 should be linear with a slope of kstretch. Table 3.1 and Fig. 3.7 show the variation of the energy Table 3.1 Change in energy as the C–C bond in CH3–CH3 is stretched away from its equilibrium ˚ length. The calculations are ab initio (STO-3G; Chapter 5). Bond lengths are in A 2 (l  leq) Estretch, kJ mol1 C–C length, l l  leq 1.538 0 0 0 1.550 0.012 0.00014 0.29 1.560 0.022 0.00048 0.89 1.570 0.032 0.00102 1.86 1.580 0.042 0.00176 3.15 1.590 0.052 0.00270 4.75 1.600 0.062 0.00384 6.67

54

3 Molecular Mechanics E stretch, kJ mol–1

7 6 5 4 3 2 1 0 0.001

0.002

0.003

2 2 0.004 (l – l eq) , Å

Fig. 3.7 Energy vs. the square of the extension of the C–C bond in CH3–CH3. The data in Table 3.1 were used

of ethane with stretching of the C/C bond, as calculated by the ab initio STO-3G method. The equilibrium bond length has been taken as the STO-3G length: leq ðC  CÞ ¼ 1:538 Å

(3.10)

The slope of the graph is kstretch (C  C) = 1,735 kJ mol1 Å

2

(3.11)

Similarly, the CH bond of methane was stretched using ab initio STO-3G calculations; the results are leq0 ðC  H) = 1:083Å kstretch (C  H)1.934 kJ mol1 Å

(3.12) 2

(3.13)

Parameterizing the Angle Bending Term From Eq. 3.3, a plot of Ebend against (a–aeq)2 should be linear with a slope of kbend. From STO-3G calculations on bending the H–C–C angle in ethane we get (cf. Table 3.1 and Fig. 3.7) aeq ðHCCÞ ¼ 110:7

(3.14)

kbend ðHCCÞ ¼ 0:093 kJ mol1 deg2

(3.15)

3.2 The Basic Principles of Molecular Mechanics

55

Table 3.2 The experimental potential energy values for rotation about the central C–C bond of CH3CH2–CH2CH3 can be approximated by Etorsion ðCH3 CH2  CH2 CH3 Þ ¼ k0 þ 4 P kr ½1 þ cosðryÞ with k0 ¼ 20.1, k1 ¼ 4.7, k2 ¼ 1.91, k3 ¼ 7.75, k4 ¼ 0.58. Experimental r¼1

energy values at 30 , 90 , and 150 were interpolated from those at 0 , 60 , 120 , and 180 ; energies are in kJ mol1 y (deg) E (calculated) E (experimental) 0 0.15 0 30 6.7 7.0 60 14 14 90 8.8 9.0 120 3.5 3.3 150 15 15 180 25 25

Calculations on staggered butane gave for the C–C–C angle aeq ðCCCÞ ¼ 112:5

(3.16)

kbend ðCCCÞ ¼ 0:110 KJ mol1 deg2

(3.17)

Parameterizing the Torsional Term For the ethane case (Fig. 3.4), the equation for energy as a function of dihedral angle can be deduced fairly simply by adjusting the basic equation E ¼ cos y to give E ¼ 1/2Emax[1 + cos3(y + 60)]. For butane (Fig. 3.5), using Eq. 3.4 and experimenting with a curve-fitting program shows that a reasonably accurate torsional potential energy function can be created with five parameters, k0 and k1–k4: Etorsion ðCH3 CH2  CH2 CH3 Þ ¼ k0 þ

4 X

kr ½1 þ cosðryÞ

(3.18)

r¼1

The values of the parameters k0–k5 are given in Table 3.2. The calculated curve can be made to match the experimental one as closely as desired by using more terms (Fourier analysis). Parameterizing the Nonbonded Interactions Term To parameterize Eq. 3.5 we might perform ab initio calculations in which the separation of two atoms or groups in different molecules (to avoid the complication of concomitant changes in bond lengths and angles) is varied, and fit Eq. 3.5 to the energy vs. distance results. For nonpolar groups this would require quite high-level calculations (Chapter 5), as van der Waals or dispersion forces are involved. We shall approximate the nonbonded interactions of methyl groups by the interactions of methane molecules, using experimental values of knb and s, derived from studies of the viscosity or the compressibility of methane. The two methods give slightly different values [7b], but we can use the values knb ¼ 4:7 kJ mol1

(3.19)

56

3 Molecular Mechanics

and s ¼ 3:85 Å

(3.20)

Summary of the Parameterization of the Forcefield Terms Eq. 3.1 were parameterized to give:

The four terms of

Estretch ðC  C) ¼ 1735ðl  1:538Þ2

(3.21)

Estretch ðC  HÞ ¼ 1934ðl  1:083Þ2

(3.22)

Ebend ðHCHÞ ¼ 0:093ða  110:7Þ2

(3.23)

Ebend ðCCCÞ ¼ 0:110ða  112:5Þ2

(3.24)

Etorsion ðCH3 CCCH3 Þ ¼ k0 þ

4 X

kr ½1 þ cosðryÞ

(3.25)

r¼1

The parameters k of Eq. 3.25 are given in Table 3.2. "   # 3:85 12 3:85 6  Enonbond ðCH3 =CH3 Þ ¼ 4:7 r r

(3.26)

Note that this parameterization is only illustrative of the principles involved; any really viable forcefield would actually be much more sophisticated. The kind we have developed here might at the very best give crude estimates of the energies of alkanes. An accurate, practical forcefield would be parameterized as a best fit to many experimental and/or calculational results, and would have different parameters for different kinds of bonds, e.g. C–C for acyclic alkanes, for cyclobutane and for cyclopropane. A forcefield able to handle not only hydrocarbons would obviously need parameters involving elements other than hydrogen and carbon. Practical forcefields also have different parameters for various atom types, like sp3 carbon vs. sp2 carbon, or amine nitrogen vs. amide nitrogen. In other words, a different value would be used for, say, stretching involving an sp3/sp3 C–C bond than for an sp2/sp2 C–C bond. This is clearly necessary since the force constant of a bond depends on the hybridization of the atoms involved; the IR stretch frequency for the sp3C/sp3C bond comes at roughly 1,200 cm1, while that for the sp2C/sp2C bond is about 1,650 cm1 [8]. Since the vibrational frequency of a bond is proportional to the square root of the force constant, the force constants are in the ratio of about (1,650/1,200)2 ¼ 1.9; for corresponding atoms, force constants are in fact generally roughly proportional to bond order (double bonds and triple bonds are about two and three times as stiff, respectively, as the corresponding single bonds). Some forcefields account for the variation of bond order with conformation

3.2 The Basic Principles of Molecular Mechanics

57

(twisting p orbitals out of alignment reduces their overlap) by performing a simple PPP molecular orbital calculation (Chapter 6) to obtain the bond order. A sophisticated forcefield might also consider H/H nonbonded interactions explicitly, rather than simply subsuming them into methyl/methyl interactions (combining atoms into groups is the feature of a united atom forcefield). Furthermore, nonbonding interactions between polar groups need to be accounted for in a field not limited to hydrocarbons. These are usually handled by the well-known potential energy/electrostatic charge relationship E ¼ kðq1 q2 =rÞ which has also been used to model hydrogen bonding [11]. A subtler problem with the naive forcefield developed here is that stretching, bending, torsional and nonbonded terms are not completely independent. For example, the butane torsional potential energy curve (Fig. 3.5) does not apply precisely to all CH3–C–C–CH3 systems, because the barrier heights will vary with the length of the central C–C bond, obviously decreasing (other things being equal) as the bond is lengthened, since there will be a decrease in the interactions (whatever causes them) between the CH3’s and H’s on one of the carbons of the central C–C and those on the other carbon. This could be accounted for by making the k’s of Eq. 3.25 a function of the X–Y length. Actually, partitioning the energy of a molecule into stretching, bending, etc. terms is somewhat formal; for example, the torsional barrier in butane can be considered to be partly due to nonbonded interactions between the methyl groups. It should be realized that there is no one, right functional form for an MM forcefield (see, e.g., [1b]); accuracy, versatility and speed of computation are the deciding factors in devising a forcefield.

3.2.3

A Calculation Using Our Forcefield

Let us apply the naive forcefield developed here to comparing the energies of two 2,2,3,3-tetramethylbutane ((CH3)3CC(CH3)3, i.e. t-Bu-Bu-t) geometries. We compare the energy of structure 1 (Fig. 3.8) with all the bond lengths and angles at our “natural” or standard values (i.e. at the STO-3G values we took as the equilibrium bond lengths and angles in Section 3.2.2) with that of structure 2, where the central ˚ to 1.600 A ˚ , but all other bond lengths, as C-C bond has been stretched from 1.538 A well as the bond angles and dihedral angles, are unchanged. Figure 3.8 shows the nonbonded distances we need, which would be calculated by the program from bond lengths, angles and dihedrals. Using Eq. 3.1: E¼

X bonds

Estretch þ

X angles

Ebend þ

X dihedrals

Etorsion þ

X pairs

! Enonbond

58

3 Molecular Mechanics 3.065 3.120

H3C 3.931 C

H3C

CH3 CH3 C CH3

H3C

H 3C

keeping bond angles and other bond lengths constant

CH3 CH3

3.974

stretch central C-C bond

C

C

H3C

CH3

H3C

1.538 Å

1.600 Å

1

2

Fig. 3.8 Structures for a simple MM “by hand” calculation on the effect of changing the central ˚ to 1.600 A ˚ C–C length of (CH3)3C–C(CH3)3 from 1.538 A

For structure 1 X

Estretch ðC  CÞ ¼ 7  1; 735 ð1:538  1:538Þ2 ¼ 0

bonds

Bond stretch contribution cf: structure with leq ¼ 1:538 X

Estretch ðC  HÞ ¼ 18  1934 ð1:083  1:083Þ2 ¼ 0

bonds

Bond stretch contribution cf: structure with leq ¼ 1:083 X

Ebend ðHCHÞ ¼ 18  0:093ð110:7  110:7Þ2 ¼ 0

angles

Bond bend contribution cf: structure with aeq ¼ 110:70 X

Ebend ðCCCÞ ¼ 12  0:110 ð112:5  112:5Þ2 ¼ 0

angles

Bond bend contribution cf: structure with aeq ¼ 112:50 X

Etorsion ðCH3 CCCH3 Þ ¼ 6  3:5 ¼ 21:0

dihedrals

Torsional contribution cf: structure with no gauche  butane interactions Actually, nonbonding interactions are already included in the torsional term (as gauche–butane interactions); we might have used an ethane-type torsional function and accounted for CH3/CH3 interactions entirely with nonbonded terms. However, in comparing calculated relative energies the torsional term will cancel out.

3.2 The Basic Principles of Molecular Mechanics

59

P

P Enonbond ðanti  CH3 =CH3 Þ þ Enonbond ðgauche  CH3 =CH3 Þ nonbond nonbond " "   #    # 3:85 12 3:85 6 3:85 12 3:85 6   ¼ 3  4:7 þ 6  4:7 3:931 3:931 3:065 3:065 ¼ 3  ð0:487Þ þ 6  ð54:05Þ ¼ 1:463 þ 324:3 ¼ 323 kJ mol1 nonbonding contribution cf: structure with noninteracting CH=3 S Etotal ¼ Estretch þ Ebend þ Etorsion ¼ 0 þ 0 þ 21:0 þ 323 kJ mol1 ¼ 344 kJ mol1 For structure 2 X Estretch ðC  CÞ ¼ 6  1735ð1:538  1:538Þ2 þ 1  1; 735ð1:600  1:538Þ2 bonds

¼ 0 þ 6:67 ¼ 6:67 kJ mol1 Bond stretch contribution cf: structure with leq ¼ 1:538 X Estretch ðC  HÞ ¼ 18  1; 934ð1:083  1:083Þ2 ¼ 0 bonds

Bond stretch contribution cf: structure with leq ¼ 1:083 X Ebend ðHCHÞ ¼ 18  0:093ð110:7  110:7Þ2 ¼ 0 angles

Bond bend contribution cf: structure with aeq ¼ 110:70 X Ebend ðCCCÞ ¼ 12  0:110ð112:5  112:5Þ2 ¼ 0 angles

Bond bend contribution cf: structure with aeq ¼ 112:50 X Etorsion ðCH3 CCCH3 Þ ¼ 6  3:5 ¼ 21:0 dihedrals

Torsional contribution cf: structure with no gauche  butane interactions The stretching and bending terms for structure 2 are the same as for structure 1, except for the contribution of the central C-C bond; strictly speaking, the torsional term should be smaller, since the opposing C(CH3) groups have been moved apart. X

Enonbond ðanti  CH3 /CH3 Þ þ

nonbond

¼ 3  4:7

"

3:85 3:974

12



3:85  3:974

X

Enonbond ðgauche  CH3 =CH3 Þ

nonbond

6 #

"

þ 6  4:7

3:85 3:120

12



3:85  3:120

¼ 3  ð0:673Þ þ 6  ð41:97Þ ¼ 2:019 þ 251:8 ¼ 250kJ mol1 nonbonding contribution cf: structure with noninteracting CH=3 s

6 #

60

3 Molecular Mechanics

Etotal ¼ Estretch þ Ebend þ Etorsion ¼ 6:67 þ 0 þ 21:0 þ 250 kJ mol1 ¼ 277 kJ mol1 So the relative energies are calculated to be Eðstructure 2Þ  Eðstructure 1Þ ¼ 277  344 kJ mol1 ¼ 67 kJ mol1 This crude method predicts that stretching the central C/C bond of 2,2,3,3-tetramethylbutane from the approximately normal sp3–C–sp3–C length of ˚ (structure 1) to the quite “unnatural” length of 1.600 A ˚ (structure 2) will 1.583 A 1 lower the potential energy by 67 kJ mol , and indicates that the drop in energy is due very largely to the relief of nonbonded interactions. A calculation using the accurate forcefield MM3 [12] gave an energy difference of 54 kJ mol1 between a “standard” geometry approximately like structure 1, and a fully optimized ˚ . The surprisingly geometry, which had a central C/C bond length of 1.576 A good agreement is largely the result of a fortuitous cancellation of errors, but this does not gainsay the fact that we have used our forcefield to calculate something of chemical interest, namely the relative energy of two molecular geometries. In principle, we could have found the minimum-energy geometry according to this forcefield, i.e. we could have optimized the geometry (Chapter 2). Geometry optimization is in fact the main use of MM, and modern programs employ analytical first and second derivatives of the energy with respect to the geometric coordinates for this (Chapter 2).

3.3

Examples of the Use of Molecular Mechanics

If we consider the applications of MM from the viewpoint of the goals of those who use it, then the main applications have been: 1. To obtain reasonable input geometries for lengthier (ab initio, semiempirical or density functional) kinds of calculations. 2. To obtain good geometries (and perhaps energies) for small- to medium-sized molecules. 3. To calculate the geometries and energies of very large molecules, usually polymeric biomolecules (proteins and nucleic acids). 4. To generate the potential energy function under which molecules move, for molecular dynamics or Monte Carlo calculations. 5. As a (usually quick) guide to the feasibility of, or likely outcome of, reactions in organic synthesis. Examples of these five facets of the use of MM will be given.

3.3 Examples of the Use of Molecular Mechanics

3.3.1

61

To Obtain Reasonable Input Geometries for Lengthier (Ab Initio, Semiempirical or Density Functional) Kinds of Calculations

The most frequent use of MM is probably to obtain reasonable starting structures for ab initio, semiempirical, or DFT (Chapters 5, 6 and 7) calculations. Nowadays this is usually done by building the molecule with an interactive builder in a graphical user interface, with which the molecule is assembled by clicking atoms or groups together, much as one does with a “real” model kit. A click of the mouse then invokes MM and provides, in most cases, a reasonable geometry. The resulting MM-optimized structure is then subjected to an ab initio, etc. calculation. By far the main use of MM is to find reasonable geometries for “normal” molecules, but it has also been used to investigate transition states. The calculation of transition states involved in conformational changes is a fairly straightforward application of MM, since “reactions” like the interconversion of butane or cyclohexane conformers do not involve the deep electronic reorganization that we call bond-making or bond-breaking. The changes in torsional and nonbonded interactions that accompany them are the kinds of processes that MM was designed to model, and so good transition state geometries and energies can be expected for this particular kind of process; transition state geometries cannot be (readily) measured, but the MM energies for conformational changes agree well with experiment: indeed, one of the two very first applications of MM [3a, d] was to the rotational barrier in biphenyls (the other was to the SN2 reaction [3c]). Since MM programs are usually not able to optimize an input geometry toward a saddle point (see below), one normally optimizes to a minimum subject to the symmetry constraint expected for the transition state. Thus for ethane, optimization to a minimum within D3h symmetry (i.e. by constraining the HCCH dihedral to be 0 , or by starting with a structure of exactly D3h symmetry) will give the transition state, while optimization with D3d symmetry gives the ground-state conformer (Fig. 3.9). Optimizing an input C2v cyclohexane structure (Fig. 3.10) gives the stationary point nearest this input structure, which is the transition state for interconversion of enantiomeric twist cyclohexane conformers. There are several examples of the application of MM to actual chemical reactions, as distinct from conformational changes; the ones mentioned here are taken from the review by Eksterowicz and Houk [13]. The simplest way to apply MM to transition states is to approximate the transition state by a ground-state molecule. This can sometimes give surprisingly good results. The rates of solvolysis of compounds RX to the cation correlated well with the energy difference between the hydrocarbon RH, which approximates RX, and the cation Rþ, which approximates the transition state leading to this cation. This is not entirely unexpected, as the Hammond postulate [14] suggests that the transition state should resemble the cation. In a similar vein, the activation energy for solvolysis has been approximated as the energy difference between a “methylalkane”, with CH3 corresponding to X in

62

3 Molecular Mechanics Input structure with this symmetry will be optimized to the transition state

energy D3h

Input structures not of D3h symmetry will be optimized to the minimum-energy conformation

dihedral angle

Fig. 3.9 Optimizing ethane within D3h symmetry (i.e. by constraining the HCCH dihedral to be 0 , or by inputting a structure with exact D3h symmetry) will give the transition state, while optimization without requiring D3d symmetry gives the ground-state conformer

energy Cs halfchair

A C2v input structure will be optimized to the transition state linking the C2 conformers C2v boat

C2 twist, or twist-boat

D3d chair reaction coordinate

Fig. 3.10 Optimizing cyclohexane within C2v symmetry gives a transition state, not one of the minima

RX, and a ketone, the sp2 carbon of which corresponds to the incipient cationic carbon of the transition state. One may wish a more precise approximation to the transition state geometry than is represented by an intermediate or a compound somewhat resembling the transition state. This can sometimes be achieved by optimizing to a minimum, subject to the constraint that the bonds being made and broken have lengths

3.3 Examples of the Use of Molecular Mechanics 1

63

2

3

H

H2C

H

C optimize C2v

start with chair cyclohexane

attach a CH2 1,4

4

5 H

H C

6 H

H

H

C

constrain to (2.1 Å)

constrain to (2.1 Å)

2

H H 1 4

H 3 optimize

optimize

2.1 Å Cs

2.1 Å

2.1 Å make a C / C double bond; set constraints on two C / C bonds

2.1 Å

remove CH2, set 1, 2, 3, 4 dihedral to 0

7 H

H H

H H

butadiene

H H

H Cs

2.1 Å 2.1 Å

Cf. the transition state

H

from

H H

H

H ethene H

Fig. 3.11 Using molecular mechanics to get the (approximate) transition state for the Diels–Alder reaction of butadiene with ethene. This procedure gives a structure with the desirable Cs, rather than a lower, symmetry

believed (e.g. from quantum mechanical calculations on simple systems, or from chemical intuition) to approximate those in the transition state, and with appropriate angles and dihedrals also constrained. With luck this will take the input structure to a point on the potential energy surface near the saddle point. For example, an approximation to the geometry of the transition state for formation of cyclohexene in the Diels–Alder reaction of butadiene with ethene can be achieved (Fig. 3.11) by essentially building a boat conformation of cyclohexene, constraining the two ˚ , and optimizing, using the CH2 bridge (later forming C/C bonds to about 2.1 A removed) to avoid twisting and to maintain Cs symmetry; optimization with a dihedral constraint removes steric conflict between two hydrogens and gives a reasonable starting structure for, say, an ab initio optimization. The most sophisticated approach to locating a transition state with MM would be to use an algorithm that optimizes the input structure to a true saddle point, that is to a geometry characterized by a Hessian with one and only one negative eigenvalue (Chapter 2). To do this the MM program would have to be able not only to calculate second derivatives, but should also be parameterized for the partial bonds in transition states. This is a feature lacking in standard MM forcefields, which are not, in general, used to calculate transition states. MM has been used to study the transition states involved in SN2 reactions, hydroborations, cycloadditions (mainly the Diels–Alder reaction), the Cope and Claisen rearrangements, hydrogen transfer, esterification, nucleophilic addition to

64

3 Molecular Mechanics

carbonyl groups and electrophilic C/C bonds, radical addition to alkenes, aldol condensations, and various intramolecular reactions [13].

3.3.2

To Obtain Good Geometries (and Perhaps Energies) for Small- to Medium-Sized Molecules

Molecular mechanics can provide excellent geometries for small (roughly C1 to about C10) and medium-sized (roughly C11 to C100) organic molecules. It is by no means limited to organic molecules, as forcefields like SYBYL and UFF [5] have been parameterized for most of the periodic table, but the great majority of MM calculations have been done on organics, probably largely because MM was the creation of organic chemists (this is probably because the concept of geometric structure has long been central in organic chemistry). The two salient features of MM calculations on small to medium-sized molecules is that they are fast and they can be very accurate. Times required for a geometry optimization of unbranched C20H42, of C2h symmetry, with the Merck Molecular Force Field (MMFF), the semiempirical AM1 (Chapter 6) and the ab initio HF/3–21G (Chapter 5) methods, as implemented with the program SPARTAN [15], were 1.2 s, 16 s, and 57 min, respectively (on an obsolescent machine a few years ago; these times would now by shorter by a factor of at least 2). Clearly as far as speed goes there is no contest between the methods, and the edge in favor of MM increases with the size of the molecule. In fact, MM was till recently the only practical method for calculations on molecules with more than about 100 heavy atoms (in computational chemistry a heavy atom is any atom heavier than helium). Even programs not designed specifically for macromolecules will handle molecules with thousands of atoms on a good PC. MM energies can be very accurate for families of compounds for which the forcefield has been parameterized. Appropriate parameterization permits calculation of DHf0 (heat of formation, enthalpy of formation) in addition to strain energy [1f]. For the MM2 program (see below), for standard hydrocarbons DHf0 errors are usually only 0–4 kJ mol1, which is comparable to experimental error, and for oxygen containing organics the errors are only 0–8 kJ mol1 [16]; the errors in MM conformational energies are often only about 2 kJ mol1 [17]. MM geometries are usually reasonably good for small to medium-sized molecules [4, 9a, 18]; for the MM3 program (see below) the RMS error in bond lengths for cholesteryl acetate ˚ [4a]. “Bond length” is, if unqualified, somewhat imprecise, was only about 0.007 A since different methods of measurement give somewhat different values [4a, 9] (Section 5.5.1). MM geometries are routinely used as input structures for quantummechanical calculations, but in fact the MM geometry and energy are in some cases as good as or better than those from a “higher-level” calculation [19]. The benchmark MM programs for small to medium-sized molecules are probably MM3 and MM4 [4, 5]; the Merck Molecular Force Field (MMFF) [20] is likely to become very popular too, not least because of its implementation in SPARTAN [15].

3.3 Examples of the Use of Molecular Mechanics

3.3.3

65

To Calculate the Geometries and Energies of Very Large Molecules, Usually Polymeric Biomolecules (Proteins and Nucleic Acids)

Next to generating geometries and energies of small to medium-sized molecules, the main use of MM is to model polymers, mainly biopolymers (proteins, nucleic acids, polysaccharides). Forcefields have been developed specifically for this; two of the most widely-used of these are CHARMM (Chemistry at HARvard using Molecular Mechanics) [21] (the academic version; the commercial version is CHARMm) and the forcefields in the computational package AMBER (Assisted Model Building with Energy Refinement) [22]. CHARMM was designed to deal with biopolymers, mainly proteins, but has been extended to handle a range of small molecules. AMBER is perhaps the most widely used set of programs for biological polymers, being able to model proteins, nucleic acids, and carbohydrates. Programs like AMBER and CHARMM that model large molecules have been augmented with quantum mechanical methods (semiempirical [23] and even ab initio [24]) to investigate small regions where treatment of electronic processes like transition state formation may be critical. An extremely important aspect of the modelling (which is done largely with MM) of biopolymers is designing pharmacologically active molecules that can fit into active sites (the pharmacophores) of biomolecules and serve as useful drugs. For example, a molecule might be designed to bind to the active site of an enzyme and block the undesired reaction of the enzyme with some other molecule. Pharmaceutical chemists computationally craft a molecule that is sterically and electrostatically complementary to the active site, and try to dock the potential drug into the active site. The binding energy of various candidates can be compared and the most promising ones can then be synthesized, as the second step on the long road to a possible new drug. The computationally assisted design of new drugs and the study of the relationship of structure to activity (quantitative structure-activity relationships, QSAR) is one of the most active areas of computational chemistry [25].

3.3.4

To Generate the Potential Energy Function Under Which Molecules Move, for Molecular Dynamics or Monte Carlo Calculations

Programs like those in AMBER are used not only for calculating geometries and energies, but also for simulating molecular motion, i.e. for molecular dynamics [26], and for calculating the relative populations of various conformations or other geometric arrangements (e.g. solvent molecule distribution around a macromolecule) in Monte Carlo simulations [27]. In molecular dynamics Newton’s laws of motion are applied to molecules moving in a molecular mechanics forcefield, although relatively small parts of the system (system: with biological molecules

66

3 Molecular Mechanics

in particular modelling is often done not on an isolated molecule but on a molecule and its environment of solvent and ions) may be simulated with quantum mechanical methods [23, 24]. In Monte Carlo methods random numbers decide how atoms or molecules are moved to generate new conformations or geometric arrangements (“states”) which are then accepted or rejected according to some filter. Tens of thousands (or more) of states are generated, and the energy of each is calculated by MM, generating a Boltzmann distribution.

3.3.5

As a (Usually Quick) Guide to the Feasibility of, or Likely Outcome of, Reactions in Organic Synthesis

In the past 15 years or so MM has become widely used by synthetic chemists, thanks to the availability of inexpensive computers (even very modest personal computers will easily run MM programs) and user-friendly and relatively inexpensive programs [5]. Since MM can calculate the energies and geometries of ground state molecules and (within the limitations alluded to above) transition states, it can clearly be of great help in planning syntheses. To see which of two or more putative reaction paths should be favored, one might (1) use MM like a hand-held model: examine the molecule for factors like steric hindrance or proximity of reacting groups, or (2) approximate the transition states for alternative reactions using an intermediate or some other plausible proxy (cf. the treatment of solvolysis in the discussion of transition states above), or (3) attempt to calculate the energies of competing transition states (cf. the above discussion of transition state calculations). The examples given here of the use of MM in synthesis are taken from the review by Lipkowitz and Peterson [28]. In attempts to simulate the metal-binding ability of biological acyclic polyethers, the tricyclic 1 (Fig. 3.12) and a tetracyclic analogue were synthesized, using as a guide the indication from MM that these molecules resemble the cyclic polyether 18-crown-6, which binds the potassium ion; the acyclic compounds were found to be indeed comparable to the crown ether in metal-binding ability. Enediynes like 2 (Fig. 3.12) are able to undergo cyclization to a phenyl-type diradical 3, which in vivo can attack DNA; in molecules with an appropriate triggering mechanism this forms the basis of promising anticancer activity. The effect of the length of the constraining chain (i.e. of n in 2) on the activation energy was studied by MM, aiding the design of compounds (potential drugs) that were found to be more active against tumors than are naturally-occurring enediyne antibiotics. To synthesize the very strained tricyclic system of 4 (Fig. 3.12), a photochemical Wolff rearrangement was chosen when MM predicted that the skeleton of 4 should be about 109 kJ mol1 less stable than that of the available 5. Photolysis of the diazoketone 6 gave a high-energy carbene which lay above the carbon skeleton of

3.4 Geometries Calculated by MM

67

H H

O O

H

H O

1 . (CH2)n

(CH2)n

.

2

3 N2

O

H

5

O

H

H

6

4

Fig. 3.12 Some molecules (1, 2, 4) which have been synthesized with the aid of molecular mechanics

4 and so was able to undergo Wolff rearrangement ring contraction to the ketene precursor of 4. A remarkable (and apparently still unconfirmed) prediction of MM is the claim that the perhydrofullerene C60H60 should be stabler with some hydrogens inside the cage [29].

3.4

Geometries Calculated by MM

Figure 3.13 compares geometries calculated with the Merck Molecular Force Field (MMFF) with those from a reasonably high-level ab initio calculation (MP2(fc)/ 6–31G*; Chapter 5) and from experiment. The MMFF is a popular forcefield, applicable to a wide variety of molecules. Popular prejudice holds that the ab initio method is “higher” than molecular mechanics and so should give superior geometries. The set of 20 molecules in Fig. 3.13 is also used in Chapters 5, 6, and 7, to illustrate the accuracy of ab initio, semiempirical, and density functional calculations in obtaining molecular geometries. The data in Fig. 3.13 are analyzed in Table 3.3. Table 3.4 compares dihedral angles for eight molecules, which are also used in Chapters 5, 6, and 7.

68

3 Molecular Mechanics H 1.519 1.526 (1.526) H

H

O 104.0 H 103.9 (104.5)

0.969 0.969 (0.958)

1.160 1.065 1.177 1.069 (1.153) (1.065) H C N

MMFF MP2(fc) / 6-31G* Experiment C2v

H

H H H 1.094 1.100 (1.099)

1.454 1.469 C2 (1.452) 0.976 O O 0.976 96.8 (0.965) 98.6 H (100.0) H

O

H

1.225 1.221 (1.208)

110.2 109.8 (110.6)

H

0.972 0.979 (0.966) H

C2v

115.5 115.6 H (116.5)

H 1.092 1.092 (1.100) H 1.357 1.392 (1.383)

O 110.4 97.2 (96.8)

1.094 1.092 (1.099)

1.019 1.018 (1.010)

N 1.452 1.465 (1.471)

H

H

H

H

1.437 1.445 (1.442) F Cs

H

H

C2v H

H H H

1.094 1.093 H H 108.4 H (1.096) 107.7 D3d (107.8) H 1.512 1.526 H H (1.531)

1.085 1.085 H (1.085) 117.9 116.6 (117.8) 1.336 1.337 H (1.339)

0.972 0.979 (0.975)

1.066 1.066 (1.061) H 1.200 1.218 (1.203)

H D2h H

H Dih

C3v

1.201 1.220 (1.206)

H

H

F C3v

Cs

1.463 1.463 H H (1.459)

1.505 1.513 (1.507)

116.7 116.5 (117.2)

H

124.2 124.5 (124.3)

O

H

H

H

H 1.493 1.499 (1.501) H

H

Cs

C2v

H

1.230 1.228 (1.222)

H

111.7 112.3 (112.4) H

1.339 1.338 (1.318)

110.6 115.4 (113.9)

107.1 107.4 1.093 (108.0) H H 1.097 (1.094) H 0.972 O 0.970 1.093 C (0.963) s 1.090 1.416 (1.094) 1.425 H 109.0 (1.421) 106.3 (107.2)

1.102 1.104 (1.116)

H Civ

H

1.092 1.089 (1.096)

109.9 110.0 (110.0) H

1.767 1.779 (1.781)

1.861 1.717 (1.690)

1.341 1.340 (1.336) H

O

110.4 102.6 (102.5)

Cl

Cs

Cl C3v

S 93.4 H C 2v 93.3 (92.1)

96.6 96.8 H 1.093 (96.5) H 1.090 H 1.341 (1.091) 1.093 S 1.341 1.091 1.804 (1.336) (1.091) 1.816 H (1.819) 1.809 1.809 (1.799) Me 95.8 95.8 (96.6)

Cs

1.500 1.512 (1.485) S

Me

O 107.5 107.4 (106.7)

Cs

(MMFF S / O single bond 1.806)

Fig. 3.13 A comparison of some MMFF, MP2(fc)/6–31G* and experimental geometries. Calculations are by the author and experimental geometries are from ref. [30a]. Note that all CH ˚ , all other bonds range from ca. 1.2–1.8 A ˚ , and all bond angles (except for linear bonds are ca. 1 A molecules) are ca. 90 120

This survey suggests that: For common organic molecules the Merck Molecular Force Field is nearly as good as the ab initio MP2(fc)/6–31G* method for calculating geometries. Both

Table 3.3 Error in MMFF molecular mechanics and MP2(fc)/6–31G* bond lengths and angles (From Fig. 3.13). Errors are given as MMFF/MP2. In some cases (e.g. MeOH) errors for two bonds are given, on one line and on the line below. A minus sign means that the calculated value is less than the experimental. The numbers of positive, negative, and zero deviations from experiment are summarized at the bottom of each column. The averages at the bottom of each column are arithmetic means of the absolute values of the errors ˚) Bond angle errors, a-aexp C–H Bond length errors, r-rexp (A O–H, N–H, S–H C–C C–O, N, F, Cl, S Angles Me2CO MeOH H2O (HOH) MeOH H2O 0.001/0.004 0.011/0.011 0.002/0.006 0.005/0.007 0.5/0.6 0.001/0.003 CH3CH3 HCHO H2O2 (HOO) HCHO H2O2 0.014/0.012 0.011/0.011 0.019/0.005 0.017/0.013 3.2/1.4 MeF MeOH (HCO) MeF MeOH CH2CH2 0.008/0.008 0.009/0.007 0.013/0.017/0.002 0.008/0.009 1.8/0.9 (COH) 0.9/0.6 HCN HOF HCCH HCN HCHO (HCH) 0.000/0.004 0.006/0.013 0.003/0.015 0.007/0.024 1.0/0.9 MeNH2 CH3CH2CH3 MeNH2 MeF (HCH) MeNH2 0.005/0.001 0.009/0.008 0.007/0.000 0.019/0.006 0.4/0.8 0.005/0.007 HOCl CH2CHCH3 Me2CO HOF (HOF) CH3CH3 0.002/0.003 0.003/0.004 0.008/0.002 0.008/0.006 13.6/0.4 0.021/0.020 H2S HCCCH3 MeCl MeNH2 (HCN) CH2CH2 0.000/0.000 0.005/0.004 0.004/0.004 0.014/0.002 3.3/1.5 0.005/0.014 CHCH MeSH MeSH Me2CO (CCC) 0.005/0.005 0.005/0.005 0.015/0.003 0.5/0.8 CH3CH3 (HCH) MeCl Me2SO 0.004/0.007 0.010/0.010 0.6/0.1 MeSH CH2CH2 (HCH) 0.002/0.000 0.1/1.2 0.002/0.001 (continued)

3.4 Geometries Calculated by MM 69

3þ, 8, two 0 4þ, 7, two 0 Mean of 13: 0.004/0.004

Table 3.3 (continued) C–H

7þ, 1, none 0 8þ, 0, none 0 Mean of 8: 0.007/0.008

˚) Bond length errors, r-rexp (A O–H, N–H, S–H

2þ, 7, none 0 5þ, 3, one 0 Mean of 9: 0.009/0.008

C–C

4þ, 5, none 0 6þ, 3, none 0 Mean of 9: 0.011/0.009

Bond angle errors, a-aexp C–O, N, F, Cl, S

Angles CH3CH2CH3 (CCC) 0.7/0.1 CH2CHCH3 (CCC) 0.1/0.2 MeCl (HCH) 0.1/0.0 H2S (HSH) 1.3/1.2 MeSH (CSH) 0.1/0.3 Me2SO (CSC) 0.8/0.8 (CSO) 0.8/0.7 0.8/0.7 7þ, 11, none 0 6þ, 11, one 0 Mean of 18: 1.7/0.7

70 3 Molecular Mechanics

3.4 Geometries Calculated by MM Table 3.4 MMFF, MP2(fc)/6–31G* and experimental dihedral angles (deg) Dihedral angles Errors Molecule MMFF MP2/6–31G* Exp. HOOH 129.4 121.3 119.1 [30a] FOOF 90.7 85.8 87.5 [30b] 72.1 69.0 73 [30b] FCH2CH2F (FCCF) 65.9 FCH2CH2OH (FCCO) 53.5 60.1 64.0 [30c] (HOCC) 54.1 54.6 [30c] ClCH2CH2OH (ClCCO) 65.7 65.0 63.2 [30b] (HOCC) 56.8 64.3 58.4 [30b] ClCH2CH2F (ClCCF) 69.8 65.9 68 [30b] HSSH 84.2 90.4 90.6 [30a] FSSF 82.9 88.9 87.9 [30b]

71

10/2.2 3.2/1.7 1.0/4 1.9/3.9 1.1/0.5 2.5/1.8 1.6/5.9

1.8/2.1 6.4/0.2 5.0/1.0 Deviations: 5þ, 5/4þ, 6 mean of 10: 3.5/2.3; Errors are given in the Errors column as MMFF/MP2/6–31G*. A minus sign means that the calculated value is less than the experimental. The numbers of positive and negative deviations from experiment and the average errors (arithmetic means of the absolute values of the errors) are summarized at the bottom of the Errors column. Calculations are by the author; references to experimental measurements are given for each measurement. The AM1 and PM3 dihedrals vary by a fraction of a degree depending on the input dihedral. Some molecules have calculated minima at other dihedrals in addition to those given here, e.g. FCH2CH2F at FCCF 180 .

methods give good geometries, but while these molecular mechanics calculations all take effectively about one second, MP2 geometry optimizations on these molecules require typically a minute or so. For larger molecules where MP2 would need hours, MM calculations might still take only seconds. Note, however, that ab initio methods provide information that molecular mechanics cannot, and are far more reliable for molecules outside those of the kind used in the MM training set (Section 3.2.2). The worst MMFF bond length deviation from experiment ˚ (the C¼C bond of propene; the MP2 deviation is among the 20 molecules is 0.021 A ˚ ˚ or less. The worst bond angle error 0.020 A); most of the other errors are ca. 0.01 A  is 13.6 , for HOF, and for HOCl the deviation is 7.9 , the second worst angle error in the set. This suggests a problem for the MMFF with X–O–Halogen angles, but while for CH3OF deviation from the MP2 angle (which is likely to be close to experiment) is MMFF–MP2 ¼ 110.7  102.8 ¼ 7.9 , for CH3OCl the deviation is only 112.0  109.0 ¼ 3.0 . MMFF dihedral angles are remarkably good, considering that torsional barriers are believed to arise from subtle quantum mechanical effects. The worst dihedral angle error is 10 , for HOOH, and the second worst, 5.0 , is for the analogous HSSH. The (granted, low-level) ab initio HF/3–21G (Chapter 5) and semiempirical PM3 (Chapter 6) methods also have trouble with HOOH, predicting a dihedral

72

3 Molecular Mechanics

angle of 180 . For those dihedrals not involving OO or SS bonds, (an admittedly small selection), the MMFF errors are only ca. 1 –2 , cf. ca. 2 –6 for MP2.

3.5

Frequencies and Vibrational Spectra Calculated by MM

Any method that can calculate the energy of a molecular geometry can in principle calculate vibrational frequencies, since these can be obtained from the second derivatives of energy with respect to molecular geometry (Section 2.5), and the masses of the vibrating atoms. Some commercially available molecular mechanics programs, for example the Merck Molecular Force Field as implemented in SPARTAN [15], can calculate frequencies. Frequencies are useful (Section 2.5) (1) for characterizing a species as a minimum (no imaginary frequencies) or a transition state or higher-order saddle point (one or more imaginary frequencies), (2) for obtaining zero-point energies to correct frozen-nuclei energies (Section 2.2), and (3) for interpreting or predicting infrared spectra. 1. Characterizing a species. This is not often done with MM, because MM is used mostly to create input structures for other kinds of calculations, and to study known (often biological) molecules. Nevertheless MM can yield information on the curvature of the potential energy surface (see Chapter 2), as calculated by that particular forcefield, anyway, at the point in question. For example, the MMFF-optimized geometries of D3d (staggered) and D3h (eclipsed) ethane (Figs. 3.3, 3.4) show, respectively, no imaginary frequencies and one imaginary frequency, the latter corresponding to rotation about the C/C bond. Thus the MMFF (correctly) predicts the staggered conformation to be a minimum, and the eclipsed to be a transition state connecting successive minima along the torsional reaction coordinate. Again, calculations on cyclohexane conformations with the MMFF correctly give the boat an imaginary frequency corresponding to a twisting motion leading to the twist conformation, which latter has no imaginary frequencies (Fig. 3.10). Although helpful for characterizing conformations, particularly hydrocarbon conformations, MM is less appropriate for species in which bonds are being formed and broken. For example, the symmetrical (D3h) species in the F þ CH3–F SN2 reaction, with equivalent C/F partial bonds, is incorrectly characterized by the MMFF as a minimum rather than a transition state, and the C/C bonds are calculated to ˚ long, cf. the value of ca. 1.8 A ˚ from methods known to be be 1.289 A trustworthy for transition states. 2. Obtaining zero-point energies (ZPEs). ZPEs are essentially the sum of the energies of each normal-mode vibration. They are added to the raw energies (the frozen-nuclei energies, corresponding to the stationary points on a Born–Oppenheimer surface; Section 2.3) in accurate calculations of relative energies using ab initio (Chapter 5) or DFT (Chapter 7) methods. However,

3.6 Strengths and Weaknesses of Molecular Mechanics

73

the ZPEs used for such corrections are usually obtained from an ab initio or DFT calculation. 3. Infrared spectra. The ability to calculate the energies (cm1) and relative intensities of molecular vibrations amounts to being able to calculate infrared spectra. MM as such cannot calculate the intensities of vibrational modes, since these involve changes in dipole moments (Section 5.5.3), and dipole moment is related to electron distribution, a concept that lies outside MM. However, approximate intensities can be calculated by assigning dipole moments to bonds or charges to atoms, and such methods have been implemented in MM programs [31]. Figures 3.14, 3.15, 3.16, and 3.17 compare the experimental IR spectra (taken in the gas phase by the author) of acetone, benzene, dichloromethane and methanol with those calculated with the MMFF program and by the “higher”, computationally much more demanding, ab initio MP2(fc)/6–31G* method (Chapter 5). In Chapters 5, 6, and 7, spectra for these four molecules, calculated by ab initio, semiempirical, and density functional methods, respectively, are given. MP2 spectra seem to generally match experiment better than those from MM, but the latter method furnishes a rapid way of obtaining approximate IR spectra. For a series of related compounds, MM might be a reasonable way to quickly investigate trends in frequencies and intensities. Extensive surveys of MMFF and MM4 frequencies showed that MMFF rootmean-square errors are ca. 60 cm1, and MM4 errors 25  52 cm1 [5b].

3.6 3.6.1

Strengths and Weaknesses of Molecular Mechanics Strengths

MM is fast, as shown by the times for optimization of C20H42 in Section 3.3. The speed of MM is not always at the expense of accuracy: for the kinds of molecules for which it has been parameterized, it can rival or surpass experiment in the reliability of its results (Sections 3.3 and 3.4). MM is undemanding in its hardware requirements: MM calculations on standard personal computers are quite practical. The characteristics of speed, (frequent) accuracy and modest computer requirements have given MM a place in many modelling programs. Because of its speed and the availability of parameters for almost all the elements (Section 3.3), MM – even when it does not provide very accurate geometries – can supply reasonably good input geometries for semiempirical, ab initio or density functional calculations, and this is one of its main applications. The fairly recent ability of MM programs to calculate IR spectra with some accuracy [16, 32] may presage an important application, since frequency calculation by quantum mechanical methods usually requires considerably more time than geometry optimization). Note that MM frequencies should be calculated using the MM geometry – unfortunately, MM can’t be used as a shortcut to obtaining frequencies for a species optimized by a quantum mechanical calculation (ab initio, density

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80

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1549 1534

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Fig. 3.14 Experimental (gas phase), and MM (MMFF) and ab initio (MP2(fc)/6–31G*) calculated IR spectra of acetone

functional or semiempirical), since frequencies must be calculated using the same method used for the geometry optimization (Section 2.5).

3.6.2

Weaknesses

The possible pitfalls in using MM are discussed by Lipkowitz [33]. The weaknesses stem from the fact that it ignores electrons. The philosophy behind MM is to think of a molecule as a collection of atoms subject to forces and to use any practical mathematical treatment of these forces to express the energy in terms of the geometric parameters. By parameterization MM can “calculate” electronic properties; for example, using bond dipoles it can find a dipole moment for a molecule, and using values that have been calculated for various atom types by quantum mechanics it can assign charges to atoms. However, such results are obtained purely by analogy, and their reliability can be negated by unexpected electronic factors to

3.6 Strengths and Weaknesses of Molecular Mechanics

75

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Benzene Experimental

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689

1.1e+002

Fig. 3.15 Experimental (gas phase), and MM (MMFF) and ab initio (MP2(fc)/6–31G*) calculated IR spectra of benzene

which MM is oblivious. MM cannot provide information about the shapes and energies of molecular orbitals nor about related phenomena such as electronic spectra. Because of the severely empirical nature of MM, interpreting MM parameters in terms of traditional physical concepts is dangerous; for example, the bond-stretching and angle-bending parameters cannot rigorously be identified with spectroscopic force constants [33]; Lipkowitz suggests that the MM proportionality constants (Section 3.2.1) be called potential constants. Other dangers in using MM are: 1. Using an inappropriate forcefield: a field parameterized for one class of compounds is not likely to perform well for other classes. 2. Transferring parameters from one forcefield to another. This is usually not valid. 3. Optimizing to a stationary point that may not really be a minimum (it could be a “maximum”, a transition state), and certainly may not be a global minimum (Chapter 2). If there is reason to be concerned that a structure is not a minimum,

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80 60 40 20 1276 1261

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817

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Fig. 3.16 Experimental (gas phase), and MM (MMFF) and ab initio (MP2(fc)/6–31G*) calculated IR spectra of dichloromethane

alter it slightly by bond rotation and reoptimize; a transition state should slide down toward a nearby minimum (e.g. eclipsed ethane altered slightly from the D3h geometry and optimized goes to the staggered conformer (g. 3.9). 4. Being taken in by vendor hype: MM programs, more so than semiempirical ones and unlike ab initio or DFT programs, are ruled by empirical factors (the form of the forcefield and the parameters used in it), and vendors do not usually caution buyers about potential deficiencies. 5. Ignoring solvent and nearby ions: for polar molecules using the in vacuo structure can lead to quite wrong geometries and energies. This is particularly important for biomolecules. One way to mitigate this problem is to explicitly add solvent molecules or ions to the system, which can considerably increase the time for a calculation. Another might be to subject various plausible in vacuo-optimized conformations to single-point (no geometry optimization) calculations that simulate the effect of solvent and take the resulting energies

3.6 Strengths and Weaknesses of Molecular Mechanics

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Fig. 3.17 Experimental (gas phase), and MM (MMFF) and ab initio (MP2(fc)/6–31G*) calculated IR spectra of methanol

as being more reliable than the in vacuo ones. Solvent effects are discussed in Chapter 8 Section 8.1. 6. Lack of caution about comparing energies calculated with MM. The method calculates the energy of a molecule relative to a hypothetical strainless idealization of the molecule. Using MM to calculate the relative energy of two isomers by comparing their strain energies (the normal MM energies) is dangerous because the two strain energies are not necessarily relative to the same hypothetical unstrained species (strain energies are not an unambiguous observable [34]). This is particularly true for functional group isomers, like (CH3)2O/ CH3CH2OH and CH3COCH3/H2C¼C(OH)CH3, which have quite different atom types. For isomers consisting of the same kinds of atoms (alkanes cf. alkanes, say), and especially for conformational isomers and E/Z isomers (geometric isomers), a good MM forcefield should give strain energies which reasonably represent relative enthalpies. For example, the MMFF gives for CH3COCH3/H2C¼C(OH)CH3 strain energies of 6.9/–6.6 kJ mol–1, i.e. relative

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energies of 0/–13 kJ mol–1, but the experimental value is ca. 0/44 kJ mol–1, i.e. H2C¼C(OH)CH3 is much the higher-energy molecule. On the other hand, the MMFF yields for gauche–butane/anti-butane strain energies of –21.3/–18.0 kJ mol–1, i.e. relative energies of 0/3.3 kJ mol–1, reasonably close to the experimental value of 0/2.8 kJ mol–1. For chair (D2d), twist (D2), and boat (C2v) cyclohexane, the MMFF strain energies are –14.9, 9.9 and 13.0 kJ mol–1, i.e. relative energies of 0, 24.8 and 27.9 kJ mol–1, cf. the experimental the estimates of 0, 24 and 29 kJ mol–1. MM programs can be parameterized to give, not just strain energy, but enthalpies of formation [1f], and the use of these enthalpies should make possible energy comparisons between isomers of disparate structural kinds. Although chemists often compare stabilities of isomers using enthalpies, we should remember that equilibria are actually determined by free energies. The lowest-enthalpy isomer is not necessarily the one of lowest free energy: a higher-enthalpy molecule may have more vibrational and torsional motion (it may be springier and floppier) and thus possess more entropy and hence have a lower free energy. Free energy has an enthalpy and an entropy component, and to calculate the latter, one needs the vibrational frequencies. Programs that calculate frequencies will usually also provide entropies, and with parameterization for enthalpy this can permit the calculation of free energies. Note that the species of lowest free energy is not always the major one present: one low-energy conformation could be outnumbered by one hundred of higher energy, each demanding its share of the Boltzmann pie. 7. Assuming that the major conformation determines the product. In fact, in a mobile equilibrium the product ratio depends on the relative reactivities, not relative amounts, of the conformers (the Curtin-Hammett principle [35]). 8. Failure to exercise judgement: small energy differences (say up to 10–20 kJ mol1) mean nothing in many cases. The excellent energy results referred to in Section 3.3 can be expected only for families of molecules (usually small to medium-sized) for which the forcefield has been parameterized. Many of the above dangers can be avoided simply by performing test calculations on systems for which the results are known (experimentally, or “known” from high-level quantum mechanical calculations). Such a reality check can have salutary effects on the reliability of one’s results, and not only with reference to molecular mechanics.

3.7

Summary

This chapter explains the basic principles of molecular mechanics (MM), which rests on a view of molecules as balls held together by springs. MM began in the 1940s with attempts to analyze the rates of racemization of biphenyls and of SN2 reactions. The potential energy of a molecule can be written as the sum of terms involving bond stretching, angle bending, dihedral angles and nonbonded interactions. Giving these terms explicit mathematical forms constitutes devising a forcefield, and

References

79

giving actual numbers to the constants in the forcefield constitutes parameterizing the field. An example is given of the devising and parameterization of an MM forcefield. MM is widely used to create reasonable geometries for input to other calculations. Such calculations are fast and can be very accurate, provided that the forcefield has been carefully parameterized for the types of molecules under study. Calculations on biomolecules is a very important application of MM; the pharmaceutical industry designs new drugs with the aid of MM: for example, examining how various candidate drugs fit into the active sites of biomolecules (docking) and the related aspect of QSAR are of major importance. MM is of some limited use in calculating the geometries and energies of transition states. Organic synthesis now makes considerable use of MM, which enables chemists to estimate which products are likely to be favored and to devise more realistic routes to a target molecule than was hitherto possible. In molecular dynamics MM is used to generate the forces acting on molecules and hence to calculate their motions, and in Monte Carlo simulations MM is used to calculate the energies of the many randomly generated states. MM is fast, it can be accurate, it is undemanding of computer power, and it provides reasonable starting geometries for quantum mechanical calculations. It ignores electrons, and so can provide parameters like dipole moment only by analogy. One must be cautious about the applicability of MM parameters to the problem at hand. Stationary points from MM, even when they are relative minima, may not be global minima. Ignoring solvent effects can give erroneous results for polar molecules. MM gives strain energies, the difference of which for structurally similar isomers represent enthalpy differences; parameterization to give enthalpies of formation is possible. Strictly speaking, relative amounts of isomers depend on free energy differences. The major conformation (even when correctly identified) is not necessarily the reactive one.

References 1. General references to molecular mechanics: (a) Rappe AK, Casewit CL (1997) Molecular mechanics across chemistry. University Science Books, Sausalito, CA; website http://www. chm.colostate.edu/mmac. (b) Leach AR (2001) Molecular modelling, principles and applications, 2nd edn. Prentice Hall, Essex (UK), chapter 4. (c) Burkert U, Allinger NL (1982) Molecular mechanics, ACS Monograph 177. American Chemical Society, Washington, DC. (d) Allinger NL (1976) Calculation of molecular structures and energy by force methods. In: Gold V, Bethell D (eds) Advances in physical organic chemistry, vol 13. Academic, New York. (e) Clark T (1985) A handbook of computational chemistry. Wiley, New York. (f) Levine IN (2000) Quantum chemistry, 5th edn. Prentice-Hall, Engelwood Cliffs, NJ, pp 664–680. (g) Issue no. 7 of Chem Rev (1993), 93. (h) Conformational energies: Pettersson I, Liljefors T (1996) In: Reviews in Computational Chemistry, vol 9, Lipkowitz KB, Boyd DB (eds) VCH Weinheim. (i) Inorganic and organometallic compounds: Landis CR, Root DM, Cleveland T (1995) In: Reviews in Computational Chemistry, vol 6, Lipkowitz KB, Boyd DB (eds) VCH Weinheim. (j) Parameterization: Bowen JP, Allinger NL (1991) Reviews in Computational Chemistry, vol 6, Lipkowitz KB, Boyd DB (eds) VCH Weinheim

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2. MM history: (a) References 1. (b) Engler EM, Andose JD, Schleyer PvR (1973) J Am Chem Soc 95:8005 and references therein. (c) Molecular mechanics up to the end of 1967 is reviewed in detail in: Williams JE, Stang PJ, Schleyer PvR (1968) Ann Rev Phys Chem 19:531 3. (a) Westheimer FH, Mayer JE (1946) J Chem Phys 14:733. (b) Hill TL (1946) J Chem Phys 14:465. (c) Dostrovsky I, Hughes ED, Ingold CK (1946) J Chem Soc 173. (d) Westheimer FH (1947) J Chem Phys 15:252 4. (a) Ma B, Lii J-H, Chen K, Allinger NL (1997) J Am Chem Soc 119:2570 and references therein. (b) In an MM4 study of amines, agreement with experiment was generally good: Chen K-H, Lii J-H, Allinger NL (2007) J Comp Chem 28:2391. (c) Five papers, using MM4, on Alcohols, ethers, carbohydrates, and related compounds. J Comp Chem 24 (2003): Allinger NL, Chen K-H, Lii J-H, Durkin KA, 1447; Lii J-H, Chen K-H, Durkin A, Allinger NL, 1473; Lii J-H, Chen K-H, Grindley TB, Allinger NL, 1490; Lii J-H, Chen K-H, Allinger NL, 1504; (2004) J Phys Chem A 108; Lii J-H, Chen K-H, Allinger NL, 3006 5. (a) Information on and references to molecular mechanics programs may be found in references 1 and the website of reference 1a. (b) For papers on the popular Merck Molecular Force Field and the MM4 forcefield (and information on some others) see the issue of J Comp Chem 17 (1996) 6. The force constant is defined as the proportionality constant in the equation force ¼ k x extension(of length or angle), so integrating force with respect to extension to get the energy (¼ force x extension) needed to stretch the bond gives E ¼ (k/2)(extension)2, i.e. k ¼ force constant ¼ 2kstretch (or 2kbend) 7. (a) A brief discussion and some parameters: Atkins PW (1994) Physical chemistry, 5th edn. Freeman, New York, pp 772, 773; it is pointed out here that er/s is actually a much better representation of the compressive potential than is r12. (b) Moore WJ (1972) Physical chemistry, 4th edn. Prentice-Hall, New Jersey, p 158 (From Hirschfelder JO, Curtis CF, Bird RB (1954) Molecular theory of gases and liquids. Wiley, New York). Note that our knb is called 4e here and must be multiplied by 8.31/1000 to convert it to our units of kJ mol1 8. Infrared spectroscopy: Silverstein RM, Bassler GC, Morrill TC (1981) Spectrometric identification of organic compounds, 4th edn. Wiley, New York, Chapter 3 9. Different methods of structure determination give somewhat different results; this is discussed in reference 4(a) and in: (a) Ma B, Lii J-H, Schaefer HF, Allinger NL (1996) J Phys Chem 100:8763 and (b) Domenicano A, Hargittai I (eds) (1992) Accurate molecular structures. Oxford Science, New York 10. To properly parameterize a molecular mechanics forcefield by calculations only high-level ab initio (or density functional) calculations would actually be used, but this does not affect the principle being demonstrated 11. Reference 1b, section 4.8 12. MM3: Allinger NL, Yuh YH, Lii J-H (1989) J Am Chem Soc 111:8551. The calculation was performed with MM3 as implemented in Spartan (reference 15) 13. Eksterowicz JE, Houk KN (1993) Chem Rev 93:2439 14. Smith MB, March J (2000) March’s advanced organic chemistry. Wiley, New York, pp 284–285 15. Spartan is an integrated molecular mechanics, ab initio and semiempirical program with an input/output graphical interface. It is available in UNIX workstation and PC versions: Wavefunction Inc., http://www.wavefun.com, 18401 Von Karman, Suite 370, Irvine CA 92715, USA. As of mid-2009, the latest version of Spartan was Spartan 09. The name arises from the simple or “Spartan” user interface. 16. Ref. [1c], pp 182–184 17. Gundertofte K, Liljefors T, Norby P-O, Pettersson I (1996) J Comp Chem 17:429 18. Comparison of various forcefields for geometry (and vibrational frequencies): Halgren TA (1996) J Comp Chem 17:553

References

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19. The Merck force field (ref. [18]) often gives geometries that are satisfactory for energy calculations (i.e. for single-point energies, no geometry optimization with the quantum mechanical method) with quantum mechanical methods; this could be very useful for large molecules: Hehre WJ, Yu J, Klunzinger PE (1997) A guide to molecular mechanics and molecular orbital calculations in Spartan. Wavefunction Inc., Irvine, CA, chapter 4 20. Halgren TA (1996) J Comp Chem 17:490 21. Nicklaus MC (1997) J Comp Chem 18:1056; the difference between CHARMM and CHARMm is explained here 22. (a) www.amber.ucsf.edu/amber/amber.html (b) Cornell WD, Cieplak P, Bayly CI, Gould IR, Merz KM, Ferguson DM, Spellmeyer DC, Fox T, Caldwell JW, Kollman PA (1995) J Am Chem Soc 117:5179 (c) Barone V, Capecchi G, Brunel Y, Andries M-LD, Subra R (1997) J Comp Chem 18:1720 23. (a) Field MJ, Bash PA, Karplus M (1990) J Comp Chem 11:700. (b) Bash PA, Field MJ, Karplus M (1987) J Am Chem Soc 109: 8092 24. Singh UC, Kollman P (1986) J Comp Chem 7:718 25. (a) Reference 1b, chapter 10. (b) H€ oltje H-D, Folkers G (1996) Molecular modelling, applications in medicinal chemistry. VCH, Weinheim, Germany. (c) van de Waterbeemd H, Testa B, Folkers G (eds) (1997) Computer-assisted lead finding and optimization. VCH, Weinheim, Germany 26. (a) Reference 1b, chapter 6. (b) Karplus M, Putsch GA (1990) Nature 347:631. (c) Brooks III CL, Case DA (1993) Chem Rev 93:2487. (d) Eichinger M, Grubm€ uller H, Heller H, Tavan P (1997) J Comp Chem 18:1729. (e) Marlow GE, Perkyns JS, Pettitt BM (1993) Chem Rev 93:2503. (f) Aqvist J, Warshel A (1993) Chem Rev 93:2523 27. Reference 1b, pp 410–456 28. Lipkowitz KB, Peterson MA (1993) Chem Rev 93:2463 29. (a) Saunders M (1991) Science 253:330. (b) See too Dodziuk H, Lukin O, Nowinski KS (1999) Polish J Chem 73:299 30. (a) Hehre WJ, Radom L, Schleyer pvR, Pople JA (1986) Ab initio Molecular Orbital Theory. Wiley, New York (b) M.D. Harmony, V.W. Laurie, R.L. Kuczkowski, R.H. Schwenderman, D.A. Ramsay, F.J. Lovas, W.H. Lafferty, A.G. Makai, Molecular Structures of Gas-Phase Polyatomic Molecules Determined by Spectroscopic Methods”, J. Physical and Chemical Reference Data, 1979, 8, 619–721 (c) Huang J, Hedberg K (1989) J Am Chem Soc 111:6909 31. Lii J-H, Allinger NL (1992) J Comp Chem 13:1138. The MM3 program can be bought from Tripos Associates of St. Louis, Missouri 32. Nevins N, Allinger NL (1996) J Comp Chem 17:730 and references therein 33. Lipkowitz KB (1995) J Chem Ed 72:1070 34. (a) Wiberg K (1996) Angew Chem Int Ed Engl 25:312. (b) Issue no. 5 of Chem Rev, 1989, 89. (c) Inagaki S, Ishitani Y, Kakefu T (1994) J Am Chem Soc 116:5954. (d) Nagase S (1995) Acc Chem Res 28:469. (e) Gronert S, Lee JM (1995) J Org Chem 60:6731. (f) Sella A, Basch H, Hoz S (1996) J Am Chem Soc 118:416. (g) Grime S (1996) J Am Chem Soc 118:1529. (h) Balaji V, Michl J (1988) Pure Appl Chem 60:189. (i) Wiberg KB, Ochterski JW (1997) J Comp Chem 18:108 35. Seeman JI (1983) Chem Rev 83:83 Added in press: 36. Handley CM, Popelier PLA (2010) The use of neural networks used to devise forcefields. J Phys Chem A 114:3371

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Easier Questions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

What is the basic idea behind molecular mechanics? What is a forcefield? What are the two basic approaches to parameterizing a forcefield? Why does parameterizing a forcefield for transition states present special problems? What is the main advantage of MM, generally speaking, over the other methods of calculating molecular geometries and relative energies? Why is it not valid in all cases to obtain the relative energies of isomers by comparing their MM strain energies? What class of problems cannot be dealt with by MM? Give four applications for MM. Which is the most widely used? MM can calculate the values (cm1) of vibrational frequencies, but without “outside assistance” it can’t calculate their intensities. Explain. Why is it not valid to calculate a geometry by some slower (e.g. ab initio) method, then use that geometry for a fast MM frequency calculation?

Harder Questions 1. One big advantage of molecular mechanics over other methods of calculating geometries and relative energies is speed. Does it seem likely that continued increases in computer speed could make MM obsolete? 2. Do you think it is possible (in practical terms? In principle?) to develop a forcefield that would accurately calculate the geometry of any kind of molecule? 3. What advantages or disadvantages are there to parameterizing a forcefield with the results of “high-level” calculations rather than the results of experiments? 4. Would you dispute the suggestion that no matter how accurate a set of MM results might be, they cannot provide insight into the factors affecting a chemical problem, because the “ball and springs” model is unphysical? 5. Would you agree that hydrogen bonds (e.g. the attraction between two water molecules) might be modelled in MM as weak covalent bonds, as strong van der Waals or dispersion forces, or as electrostatic attractions? Is any one of these three approaches to be preferred in principle? 6. Replacing small groups by “pseudoatoms” in a forcefield (e.g. CH3 by an “atom” about as big) obviously speeds up calculations. What disadvantages might accompany this simplification? 7. Why might the development of an accurate and versatile forcefield for inorganic molecules be more of a challenge than for organic molecules?

Harder Questions

83

8. What factor(s) might cause an electronic structure calculation (e.g. ab initio or DFT) to give geometries or relative energies very different from those obtained from MM? 9. Compile a list of molecular characteristics/properties that cannot be calculated purely by MM. 10. How many parameters do you think a reasonable forcefield would need to minimize the geometry of 1,2-dichloroethane?

Chapter 4

Introduction to Quantum Mechanics in Computational Chemistry

It is by logic that we prove, but by intuition that we discover. J.H. Poincare´, ca. 1900

Abstract A historical view demystifies the subject. The focus is strongly on chemistry. The application of quantum mechanics (QM) to computational chemistry is shown by explaining the Schr€ odinger equation and showing how this equation led to the simple H€ uckel method, from which the extended H€uckel method followed. This sets the stage well for ab initio theory, in Chapter 5. QM grew out of studies of blackbody radiation and of the photoelectric effect. Besides QM, radioactivity and relativity contributed to the transition from classical to modern physics. The classical Rutherford nuclear atom, the Bohr atom, and the Schr€ odinger wave-mechanical atom are discussed. Hybridization, wavefunctions, Slater determinants and other basic concepts are explained. For obtaining eigenvectors and eigenvalues from the secular equations the elegant and simple matrix diagonalization method is explained and used. All the necessary mathematics is explained.

4.1

Perspective

Chapter 1 outlined the tools that computational chemists have at their disposal, Chapter 2 set the stage for the application of these tools to the exploration of potential energy surfaces, and Chapter 3 introduced one of these tools, molecular mechanics. In this chapter you will be introduced to quantum mechanics, and to quantum chemistry, the application of quantum mechanics to chemistry. Molecular mechanics is based on classical physics, physics before modern physics; one of the cornerstones of modern physics is quantum mechanics, and ab initio (Chapter 5), semiempirical (Chapter 6), and density functional (Chapter 7) methods belong to quantum chemistry. This chapter is designed to ease the way to an understanding of E.G. Lewars, Computational Chemistry, DOI 10.1007/978-90-481-3862-3_4, # Springer ScienceþBusiness Media B.V. 2011

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4 Introduction to Quantum Mechanics in Computational Chemistry

the role of quantum mechanics in computational chemistry. The word quantum comes from Latin (quantus, “how much?”, plural quanta) and was first used in our sense by Max Planck in 1900, as an adjective and noun, to denote the constrained quantities or amounts in which in which energy can be emitted or absorbed. Although the term quantum mechanics was apparently first used by Born (of the Born–Oppenheimer approximation, Section 2.3) in 1924, in contrast to classical mechanics, the matrix algebra and differential equation techniques that we now associate with the term were presented in 1925 and 1926 (Section 4.2.6). “Mechanics” as used in physics is traditionally the study of the behavior of bodies under the action of forces like, e.g., gravity (celestial mechanics). Molecules are made of nuclei and electrons, and quantum chemistry deals, fundamentally, with the motion of electrons under the influence of the electromagnetic force exerted by nuclear charges. An understanding of the behavior of electrons in molecules, and thus of the structures and reactions of molecules, rests on quantum mechanics and in particular on that adornment of quantum chemistry, the Schr€odinger equation. For that reason we will consider in outline the development of quantum mechanics leading up to the Schr€ odinger equation, and then the birth of quantum chemistry with (at least as far as molecules of reasonable size goes) the application of the Schr€ odinger equation to chemistry by H€ uckel. This simple H€ uckel method is currently disdained by some theoreticians, but its discussion here is justified by the fact that (1) it continues to be useful in research and (2) it “is immensely useful as a model, today . . . Because it is the model which preserves the ultimate physics, that of nodes in wave functions. It is the model which throws away absolutely everything except the last bit, the only thing that if thrown away would leave nothing. So it provides fundamental understanding.”1 A discussion of a generalization of the simple H€ uckel method, the extended H€ uckel method, sets the stage for Chapter 5. The historical approach used here, although perforce somewhat superficial, may help to ameliorate the apparent arbitrariness of certain features of quantum chemistry [1, 2]. An excellent introduction to quantum chemistry is the text by Levine [3]. Our survey of the factors that led to modern physics and quantum chemistry will follow the sequence: 1. 2. 3. 4. 5. 6.

1

The origins of quantum theory: blackbody radiation and the photoelectric effect Radioactivity (brief) Relativity (very brief) The nuclear atom The Bohr atom The wave mechanical atom and the Schr€ odinger equation

Personal communication from Professor Roald Hoffmann, 2002 February 13. See too Section 4.4.1, footnote.

4.2 The Development of Quantum Mechanics. The Schr€ odinger Equation

4.2

87

The Development of Quantum Mechanics. The Schr€odinger Equation

4.2.1

The Origins of Quantum Theory: Blackbody Radiation and the Photoelectric Effect

Three discoveries mark the transition from classical to modern physics: quantum theory, radioactivity, and relativity (Fig. 4.1). Quantum theory had its origin in the study of blackbody radiation and the photoelectric effect.

4.2.1.1

Blackbody Radiation

A blackbody is one that is a perfect absorber of radiation: it absorbs all the radiation falling on it, without reflecting any. More relevant for us, the radiation emitted by a hot blackbody depends (as far as the distribution of energy with wavelength goes) only on the temperature, not on the material the body is made of, and is thus amenable to relatively simple analysis. The sun is approximately a blackbody; in the lab a good source of blackbody radiation is a furnace with blackened insides and a small aperture for the radiation to escape. In the second half of the nineteenth century the distribution of energy with respect to wavelength that characterizes blackbody radiation was studied, in research that is associated mainly with Lummer and Pringsheim [1]. They plotted the flux DF (in modern SI units, J s1 m2) per wavelength emitted by a blackbody over a wavelength range Dl versus the wavelength, for various temperatures (Fig. 4.2): DF/Dl versus l. The result is a histogram or bar graph in which the area of each rectangle is (DF/Dl)Dl ¼ DF and represents the flux (energy per second per unit area) emitted in the wavelength range covered by that Dl; DF/Dl can be called the flux density for that particular blackbody radiation mechanics Galileo, Newton, etc.

optics Newton, Huygens, Young

classical physics (physics before 1900)

electromagnetism Faraday, Maxwell

photoelectric effect

quantum theory Planck, Einstein, etc.

radioactivity Becquerel

modern physics (after 1900)

relativity Einstein

Fig. 4.1 The discoveries marking the transition from classical to modern to physics. Although radioactivity was discovered in 1896, its understanding had to wait for relativity and quantum theory

88

4 Introduction to Quantum Mechanics in Computational Chemistry

CLASSICAL PHYSICS (FALSE)

FACT

ΔF Δλ

ΔF Δλ

λ

λ

Fig. 4.2 In the limit the bar graph becomes a curve, the graph of f(l) versus l, where DF dF f ðlÞ ¼ lim Dl ¼ dl , essentially intensity of radiation versus wavelength. Planck’s efforts to find ! Dl 0

the function f(l) led to the quantum theory

wavelength range Dl. The total area of all the rectangles is the total flux emitted over its whole wavelength range by the blackbody. As Dl approaches zero (note that for the nonmonochromatic radiation from a blackbody the flux at a particular wavelength is essentially zero) the histogram approaches being a smooth curve, the ratio of finite increments approximates a derivative, and we can ask: what is the function (Fig. 4.2) dF/dl ¼ f(l)? In the answer to this question lay the beginnings of quantum theory. Late nineteenth century physics, classical physics at its zenith, predicted that the flux density emitted by a blackbody should rise without limit as the wavelength decreases. This is because classical physics held that radiation of a particular frequency was emitted by oscillators (atoms or whatever) vibrating with that frequency, and that the average energy of an oscillator was independent of its frequency; since the number of possible frequencies increases without limit, the flux density (energy per second per unit area per wavelength interval) from the blackbody should rise without limit toward higher frequencies or shorter wavelengths, into the ultraviolet, and so the total flux (energy per second per unit area) should be infinite. This is clearly absurd and was recognized as being absurd; in fact, it was called “the ultraviolet catastrophe” [1]. To understand the nature of blackbody radiation and to escape the ultraviolet catastrophe, physicists in the 1890s tried to find the function (Fig. 4.2) f(l). Without breaking with classical physics, Wien had found a theoretical equation that fit the Lummer–Pringsheim curve at relatively short wavelengths, and Rayleigh and Jeans one that fit at relatively long wavelengths. Max Planck2 adopted a different approach: he found, in 1900, a purely empirical equation dF/dl ¼ f(l)

2

Max Planck, born Kiel, Germany, 1858. Ph.D. Berlin 1879. Professor, Kiel, Berlin. Nobel Prize in physics for quantum theory of blackbody radiation 1918. Died G€ ottingen, 1947.

4.2 The Development of Quantum Mechanics. The Schr€ odinger Equation

89

that fit the facts, and then tried to interpret the equation theoretically. To do this he had to make two assumptions: 1. The total energy possessed by the oscillators in the frequency range n þ dn (n is the Greek letter nu, commonly used for frequency, not to be confused with v, vee, commonly used for velocity) is proportional to the frequency: Etot ðn þ dnÞ / n

(4.1)

2. The emission or absorption of radiation of frequency n by the collection of oscillators is caused by jumps between energy levels, with loss or gain of a quantity of energy kn:

DE ¼ kn

(4.2)

The constant k, now recognized as a fundamental constant of nature, 6.626  1034 J s particle1, is called Planck’s constant, and is denoted by h, so Eq. 4.2 becomes DE ¼ hn

(4:3)

Why the letter h? Evidently because h is sometimes used in mathematics to denote infinitesimals and Planck intended to let this quantity go to zero (this was suggested to me by the late Professor Philip Morrison of MIT). In the event, it turned out to be small but finite. Apparently the letter was first used to denote the new constant in a talk given by Planck at a meeting (Sitzung) of the German Physical Society in Berlin, on 14 December 1900 [4]. The interpretation of Eq. 4.3, a fundamental equation of quantum theory, as meaning that the energy represented by radiation of frequency n is absorbed and emitted in quantized amounts hn (definite, constrained amounts; jerkily rather than continuously) was, ironically, apparently never fully accepted by Planck [5]. Planck’s constant is a measure of the graininess of our universe: because it is so small processes involving energy changes often seem to take place smoothly, but on an ultramicroscopic scale the graininess is there [6]. The constant h is the hallmark of quantum expressions, and its finite value distinguishes our universe from a nonquantum one.

4.2.1.2

The Photoelectric Effect

An apparently quite separate (but in science no two phenomena are really ever unrelated) phenomenon that led to Eq. 4.3, which is to say to quantum theory, is the photoelectric effect: the ejection of electrons from a metal surface exposed to light.

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4 Introduction to Quantum Mechanics in Computational Chemistry

kinetic energy of the emitted electrons, 1 / 2 mv 2

1 / 2 mv 2 = h

–W

0 frequency,

, of the light

–W

Fig. 4.3 The photoelectric effect. Einstein explained the effect by extending to light Planck’s idea of the absorption and emission of energy in discrete amounts: he postulated that light itself consisted of discrete particles

The first inkling of this phenomenon was due to Hertz,3 who in 1888 noticed that the potential needed to elicit a spark across two electrodes decreased when ultraviolet light shone on the negative electrode. Beginning in 1902, the photoelectric effect was first studied systematically by Lenard,4 who showed that the phenomenon observed by Hertz was due to electron emission. Facts (Fig. 4.3) that classical physics could not explain were the existence of a threshold frequency for electron ejection, that the kinetic energy of the electrons is linearly related to the frequency of the light, and the fact that the electron flux (electrons per unit area per second) is proportional to the intensity of the light. Classical physics predicted that the electron flux should be proportional to the light frequency, decreasing with a decrease in frequency, but without sharply falling to zero below a certain frequency, and that the kinetic energy of the electrons should be proportional to the intensity of the light, not the frequency. These facts were explained by Einstein5 in 1905 in a way that now appears very simple, but in fact relies on concepts that were at the time revolutionary. Einstein went beyond Planck and postulated that not only was the process of absorption and emission of light quantized, but that light itself was quantized, consisting in effect of particles of energy 3

Heinrich Hertz, born Hamburg, Germany, 1857. Ph.D. Berlin, 1880. Professor, Karlsruhe, Bonn. Discoverer of radio waves. Died Bonn, 1894. 4 Philipp Lenard, German physicist, born Pozsony, Austria-Hungary (now Bratislava, Slovakia), 1862. Ph.D. Heidelberg 1886. Professor, Heidelberg. Nobel Prize in physics 1905, for work on cathode rays. Lenard supported the Nazis and rejected Einstein’s theory of relativity. Died Messelhausen, Germany, 1947. 5 Albert Einstein, German–Swiss–American physicist. Born Ulm, Germany, 1879. Ph.D. Z€ urich 1905. Professor Z€urich, Prague, Berlin; Institute for Advanced Studies, Princeton, New Jersey. Nobel Prize in physics 1921 for theory of the photoelectric effect. Best known for the special (1905) and general (1915) theories of relativity. Died Princeton, 1955.

4.2 The Development of Quantum Mechanics. The Schr€ odinger Equation

Eparticle ¼ hv v ¼ frequency of the light

91

(4.4)

These particles became known as photons (the word was coined by Gilbert Lewis, ca. 1923, but his photon was not the particle of modern physics). If the energy of the photon before it removes an electron from the metal is equal to the energy required to tear the electron free of the metal, plus the kinetic energy of the free electron, then hv ¼ W þ 1=2me v 2

(4.5)

W ¼ work function of the metal, energy needed to remove an electron (with no energy left over), me ¼ mass of an electron, v ¼ velocity of electron ejected by the photon, 1=2 me v 2 ¼ kinetic energy of the free electron Rearranging Eq. 4.5: 1=2m

ev

2

¼ hv  W

(4.6)

Thus a plot of the kinetic energies of the electrons ð1=2me v 2 Þ versus the frequency n of the light should be a straight line of positive slope (h; this is one way to find Planck’s constant) intersecting the n axis at a positive value (n ¼ W/h), as experiment indeed showed (Fig. 4.3). Planck’s explanation of the blackbody radiation curves (1900 [4]) and Einstein’s explanation of the facts of the photoelectric effect (1905 [7]) indicated that the flow of energy in physical processes did not take place continuously, as had been believed, but rather jerkily, in discrete jumps, quantum by quantum. The contributions of Planck and Einstein were the signal developments marking the birth of quantum theory and the transition from classical to modern physics.

4.2.2

Radioactivity

Brief mention of radioactivity is in order because it, along with quantum mechanics and relativity, transformed classical into modern physics. Radioactivity was discovered by Becquerel in 1896. However, an understanding of how materials like uranium and radium could emit, over the years, a million times more energy than would be permitted by chemical reactions, had to await Einstein’s special theory of relativity (Section 4.2.3), which showed that a tiny, unnoticeable decrease in mass represented the release of a large amount of energy.

4.2.3

Relativity

Relativity is relevant to computational chemistry because it must often be explicitly taken into account in accurate calculations involving atoms heavier than about

92

4 Introduction to Quantum Mechanics in Computational Chemistry

chlorine or bromine (see below) and because, strictly speaking, the Schr€odinger equation, the fundamental equation of quantum chemistry, is an approximation to a relativistic equation, the Dirac6 equation. Relativity was discovered in by Einstein in 1905, when he formulated the special theory of relativity, which deals with nonaccelerated motion in the absence of significant gravitational fields (general relativity, published by Einstein in 1915, is concerned with accelerated motion and gravitation). Special relativity predicted a relationship between mass and energy, the famous E ¼ mc2 equation and, of more direct relevance to computational chemistry, showed that the mass of a particle increases with its velocity, dramatically so near the velocity of light. In heavier elements the inner electrons are moving at a significant fraction of the speed of light, and the relativistic increase in their masses affects the chemistry of these elements (actually, some physicists do not like to think in terms of rest mass and relativistic mass, but that is a controversy that need not concern us here). In computational chemistry relativistic effects on electrons are usually accounted for by what are called effective core potentials or pseudopotentials (Chapter 5, Section 5.3.3.7, and Chapter 8, Section 8.3).

4.2.4

The Nuclear Atom

The “nuclear atom” is the picture of the atom as a positive nucleus surrounded by negative electrons. Although the idea of atoms in speculative philosophy goes back to at least the time of Democritus,7 the atom as the basis of a scientifically credible theory emerges only in nineteenth century, with the rationalization by Dalton8 in 1808 of the law of definite proportions. Nevertheless, atoms were regarded by many scientists of the positivist school of Ernst Mach as being at best a convenient hypothesis, despite the success of the atomistic Maxwell–Boltzmann9 kinetic

6

Paul Adrien Maurice Dirac, born Bristol, England, 1902. Ph.D. Cambridge, 1926. Professor, Cambridge, Dublin Institute for Advanced Studies, University of Miami, Florida State University. Nobel prize in physics 1933 (shared with Schr€ odinger). Known for his mathematical elegance, for connecting relativity with quantum theory, and for predicting the existence of the positron. Died Tallahassee, Florida, 1984. 7 Democritus, Greek philosopher, born Abdera, Thrace (the eastern Balkans) ca. 470 BC. Died ca. 370 BC. 8 John Dalton, born Eaglesfield, England, 1766. Considered the founder of quantitative chemical atomic theory: law of definite proportions, pioneered determination of atomic weights. Cofounder of British Association for the Advancement of Science. Died Manchester, England, 1844. 9 Ludwig Boltzmann, born Vienna 1844. Ph.D. Vienna. Professor Graz, Vienna. Developed the kinetic theory of gases independently of Maxwell (viz., Boltzmann constant’s, k). Firm supporter of the atomic theory in opposition to Mach and Ostwald, helped develop concept of entropy (S). Died Duino, Austria (now in Italy), 1906 (suicide incurred by depression). Inscribed on gravestone: S ¼ k log W.

4.2 The Development of Quantum Mechanics. The Schr€ odinger Equation

93

theory of gases and it was not until 1908, when Perrin’s10 experiments confirmed Einstein’s atomistic analysis of Brownian motion that the reality of atoms was at last accepted by such eminent holdouts as Boltzmann’s opponent Ostwald.11 The atom has an internal structure; it is thus not “atomic” in the Greek sense and is more than the mere restless particle of kinetic theories of gases or of Brownian motion. This was shown by two lines of work: the study of the passage of electricity through gases and the behavior of certain solutions. The study of the passage of electricity through gases at low pressure was a very active field of research in the nineteenth century and only a few of the pioneers in what we can now see as the incipient field of subatomic physics will be mentioned here. The observation by Pl€ ucker in 1858 of a fluorescent glow near the cathode on the glass walls of a current-carrying evacuated tube was one of the first inklings that particles might be elicited from atoms. That these were indeed particles rather than electromagnetic rays was indicated by Crookes in the 1870s, by showing that they could be deflected by a magnet. Goldstein showed in 1886 the presence of particles of opposite charge to those emitted from the cathode, and christened the latter “cathode rays”. That the cathode rays were negative particles was proved by Perrin in 1895, when he showed that they imparted a charge to an object on which they fell. Further evidence of the particle nature of cathode rays came at around the same time from Thomson,12 who showed (1897) that they are deflected in the expected direction by an electric field. Thomson also measured their mass-to-charge ratio and from the smallest possible value of charge in electrochemistry calculated the mass of these particles to be about 1/1,837 of the mass of a hydrogen atom. Lorentz later applied the name “electron” to the particle, adopting a term that had been appropriated from the Greek by Stoney for a unit of electric current (elektron: amber, which acquires a charge when rubbed). Thomson has been called the discoverer of the electron. It was perhaps Thomson who first suggested a specific structure for the atom in terms of subatomic particles. His “plum pudding” model (ca. 1900), which placed electrons in a sea of positive charge, like raisins in a pudding., accorded with the then-known facts in evidently permitting electrons to be removed under the influence of an electric potential. The modern picture of the atom as a positive nucleus with extranuclear electrons was proposed by Rutherford13 in 1911. It arose from

10 Jean Perrin, born Lille, France, 1870. Ph.D. E´cole Normal Supe´rieure, Paris. Professor University of Paris. Nobel Prize in physics 1923. Died New York, 1942. 11 Wilhelm Friedrich Ostwald, German chemist, born Riga, Latvia, 1853. Ph.D. Dorpat, Estonia. Professor Riga, Leipzig. A founder of physical chemistry, opponent of the atomic theory till convinced by the work of Einstein and Perrin. Nobel Prize in chemistry 1909. Died near Leipzig, 1932. 12 Sir Joseph John Thomson, born near Manchester, 1856. Professor, Cambridge. Nobel Prize in physics 1906. Knighted 1908. Died Cambridge, 1940. 13 Ernest Rutherford (Baron Rutherford), born near Nelson New Zealand, 1871. Studied at Cambridge under J. J. Thomson. Professor McGill University (Montreal), Manchester, and Cambridge. Nobel prize in chemistry 1908 for work on radioactivity, alpha particles, and atomic structure. Knighted 1914. Died London, 1937.

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4 Introduction to Quantum Mechanics in Computational Chemistry

experiments in which alpha particles from a radioactive sample were shot through very thin gold foil. Most of the time the particles passed through, but occasionally one bounced back, indicating that the foil was mostly empty space, but that present were particles which were small and, compared to the mass of the electron (which was much too light to stop an alpha particle), massive. From these experiments emerged our picture of the atom as consisting of a small, relatively massive positive nucleus surrounded by electrons: the nuclear atom. Rutherford gave the name proton (from Greek protoz, primary or first) to the least massive of these nuclei (the hydrogen nucleus). There is another thread to the development of the concept of the atom as a composite of subatomic particles. The enhanced effect of electrolytes (solutes that provide electrically conducting solutions) on boiling and freezing points and on the osmotic pressure of solutions led Arrhenius14 in 1884 to propose that these substances exist in water as atoms or groups of atoms with an electric charge. Thus sodium chloride in solution would not, as was generally held, exist as NaCl molecules but rather as a positive sodium “atom” and a negative chlorine “atom”; the presence of two particles instead of the expected one accounted for the enhanced effects. The ability of atoms to lose or gain charge hinted at the existence of some kind of subatomic structure, and although the theory was not warmly received (Arrhenius was almost failed on his Ph.D. exam), the confirmation by Thomson (ca. 1900) that the atom contains electrons made acceptable the concept of charged atoms with chemical properties quite different from those of the neutral ones. Arrhenius was awarded the Nobel Prize for his (albeit significantly modified) Ph.D. work.

4.2.5

The Bohr Atom

The nuclear atom as formulated by Rutherford faced a serious problem: the electrons orbit the nucleus like planets orbiting the Sun. An object engaged in circular (or elliptical) motion experiences an acceleration because its direction is changing and thus its velocity, which unlike speed is a vector, is also changing. An electron in circular motion about a nucleus would experience an acceleration toward the nucleus, and since from Maxwell’s equations of electromagnetism an accelerated electric charge radiates away energy, the electron should lose energy by spiralling in toward the nucleus, ending up there, with no kinetic and potential energy; calculations show this should happen in a fraction of a second [8].

14

Svante Arrhenius, born near Uppsala, Sweden, 1859. Ph.D. University of Stockholm. Nobel Prize in chemistry 1903. Professor Stockholm. Died Stockholm 1927.

4.2 The Development of Quantum Mechanics. The Schr€ odinger Equation

95

A way out of this dilemma was suggested by Bohr15 in 1913 [9, 10]. He retained the classical picture of electrons orbiting the nucleus in accord with Newton’s laws, but subject to the constraint that the angular momentum of an electron must be an integral multiple of h/2p: mvr ¼ n ðh=2pÞ;

n ¼ 1; 2; 3;

(4.7)

m ¼ electron mass v ¼ electron velocity r ¼ radius of electron orbit h ¼ Planck’s constant Equation 4.7 is the Bohr postulate, that electrons can defy Maxwell’s laws provided they occupy an orbit whose angular momentum (corresponding to an orbit of appropriate radius) satisfies Eq. 4.7. The Bohr postulate is not based on a whim, as most textbooks imply, but rather follows from: (1) the Plank equation Eq. 4.3, DE ¼ hn and (2) starting with an orbit of large radius such that the motion is essentially linear and classical physics applies, as no acceleration is involved, then extrapolating to small-radius orbits. The fading of quantum-mechanical equations into their classical analogues as macroscopic conditions are approached is called the correspondence principle [11]. Using the postulate of Eq. 4.7 and classical physics, Bohr derived an equation for the energy of an orbiting electron in a one-electron atom (a hydrogenlike atom, H or Heþ, etc.) in terms of the charge on the nucleus and some constants of nature. Starting with the total energy of the electron as the sum of its kinetic and potential energies: Et ¼

1 Ze2 mv 2  2 4p e0 r

(4.8)

Z ¼ nuclear charge (1 for H, 2 for He, etc.), e ¼ electron charge, e0 ¼ permitivity of the vacuum. Using force ¼ mass  acceleration: Ze2 mv 2 ¼ 4 p e0 r 2 r

(4.9)

Ze2 ¼ mv 2 4 p e0 r

(4.10)

i.e.

15

Niels Bohr, born Copenhagen, 1885. Ph.D. University of Copenhagen. Professor, University of Copenhagen. Nobel Prize in physics 1922. Founder of the “Copenhagen school” interpretation of quantum theory. Died Copenhagen, 1962.

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4 Introduction to Quantum Mechanics in Computational Chemistry

So from Eq. 4.8 Et ¼

1 1 mv 2  mv 2 ¼  mv 2 2 2

(4.11)

Ze2 2 e0 n h

(4.12)

Z2 e 4 m 8 e0 2 n 2 h2

(4.13)

From Eqs. 4.7 and 4.10: v¼ So from Eqs. 4.11 and 4.12: Et ¼ 

Equation 4.13 expresses the total (kinetic plus potential) energy of the electron of a hydrogenlike atom in terms of four fundamental quantities of our universe: electron charge, electron mass, the permittivity of empty space, and Planck’s constant. From Eq. 4.13 the energy change involved in emission or absorption of light by a hydrogenlike atom is simply
 m Z 2 e4 1 1  DE ¼ Et2  Et1 ¼ 8 e0 2 h2 n1 2 n2 2

(4.14)

where DE is the energy of a state characterized by quantum number n2, minus the energy of a state characterized by quantum number n1. Note that from Eq. 4.13 the total energy increases (becomes less negative) as n increases (¼ 1, 2, 3, . . .), so higher-energy states are associated with higher quantum numbers n and DE > 0 corresponds to absorption of energy and DE < 0 to emission of energy. The Planck relation between the amount of radiant energy absorbed or emitted and its frequency (DE ¼ hn, Eq. 4.3), Eq. 4.14 enables one to calculate the frequencies of spectroscopic absorption and emission lines for hydrogenlike atoms. The agreement with experiment is excellent, and the same is true too for the calculated ionization energies of hydrogenlike atoms (DE for n2 ¼ 1 in Eq. 4.14).

4.2.6

The Wave Mechanical Atom and the Schr€ odinger Equation

The Bohr approach works well for hydrogenlike atoms, atoms with one electron: hydrogen, singly-ionized helium, doubly-ionized lithium, etc. However, it showed

4.2 The Development of Quantum Mechanics. The Schr€ odinger Equation

97

many deficiencies for other atoms, which is to say, almost all atoms of interest other than hydrogen. The problems with the Bohr atom for these cases were: 1. There were lines in the spectra corresponding to transitions other than simply between two n values (cf. Eq. 4.14). This was rationalized by Sommerfeld in 1915, by the hypothesis of elliptical rather than circular orbits, which essentially introduced a new quantum number k, a measure of the eccentricity of the elliptical orbit. Electrons could have the same n but different k’s, increasing the variety of possible electronic transitions; k is related to what we now call the azimuthal quantum number, l; l ¼ k  1). 2. There were lines in the spectra of the alkali metals that were not accounted for by the quantum numbers n and k. In 1925 Goudsmit and Uhlenbeck showed that these could be explained by assuming that the electron spins on an axis; the magnetic field generated by this spin around an axis could reinforce or oppose the field generated by the orbital motion of the electron around the nucleus. Thus for each n and k there are two closely-spaced “magnetic levels”, making possible new, closely-spaced spectral lines. The spin quantum number, ms ¼ þ1/2 or 1/2, was introduced to account for spin. 3. There were new lines in atomic spectra in the presence of an external magnetic field (not to be confused with the fields generated by the electron itself). This Zeeman effect (1896) was accounted for by the hypothesis that the electron orbital plane can take up only a limited number of orientations, each with a different energy, with respect to the external field. Each orientation was associated with a magnetic quantum number mm (often designated m) ¼ l, (l  1),..., (l  1), l). Thus in an external magnetic field the numbers n, k (later l) and ms are insufficient to describe the energy of an electron and new transitions, invoking mm, are possible. The only quantum number that flows naturally from the Bohr approach is the principal quantum number, n; the azimuthal quantum number l (a modified k), the spin quantum number ms and the magnetic quantum number mm are all ad hoc, improvised to meet an experimental reality. Why should electrons move in elliptical orbits that depend on the principal quantum number n? Why should electrons spin, with only two values for this spin? Why should the orbital plane of the electron take up with respect to an external magnetic field only certain orientations, which depend on the azimuthal quantum number? All four quantum numbers should follow naturally from a satisfying theory of the behaviour of electrons in atoms. The limitations of the Bohr theory arise because it does not reflect a fundamental facet of nature, namely the fact that particles possess wave properties. These limitations were transcended by the wave mechanics of Schr€odinger,16 when he devised his famous equation in 1926 [12, 13]. Actually, the year before the Schr€odinger

16

Erwin Schr€odinger, born Vienna, 1887. Ph.D. University of Vienna. Professor Stuttgart, Berlin, Graz (Austria), School for Advanced Studies Dublin, Vienna. Nobel Prize in physics 1933 (shared with Dirac). Died Vienna, 1961.

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4 Introduction to Quantum Mechanics in Computational Chemistry

equation was published, Heisenberg17 published his matrix mechanics approach to calculating atomic (and in principle molecular) properties. The matrix approach is at bottom equivalent to Schr€ odinger’s use of differential equations, but the latter has appealed to chemists more because, like physicists of the time, they were unfamiliar with matrices (Section 4.3.3), and because the wave approach lends itself to a physical picture of atoms and molecules while manipulating matrices perhaps tends to resemble numerology. Matrix mechanics and wave mechanics are usually said to mark the birth of quantum mechanics (1925, 1926), as distinct from quantum theory (1900). We can think of quantum mechanics as the rules and equations used to calculate the properties of molecules, atoms, and subatomic particles. Wave mechanics grew from the work of de Broglie,18 who in 1923 was led to this “wave-particle duality” by his ability to deduce the Wien blackbody equation (Section 4.2.1) by treating light as a collection of particles (“light quanta”) analogous to an ideal gas [14]. This suggested to de Broglie that light (traditionally considered a wave motion) and the atoms of an ideal gas were actually not fundamentally different. He derived a relationship between the wavelength of a particle and its momentum, by using the time-dilation principle of special relativity, and also from an analogy between optics and mechanics. The reasoning below, while perhaps less profound than de Broglie’s, may be more accessible. From the special theory of relativity, the relation between the energy of a photon and its mass is Ep ¼ mc2

(4:15)

where c is the velocity of light. From the Planck equation 4.3 for the emission and absorption of radiation, the energy Ep of a photon may be equated with the energy change DE of an oscillator, and we may write Ep ¼ hn

(4:16)

mc2 ¼ hn

(4.17)

From Eqs. 4.15 and 4.16

Since n ¼ c/l, Eq. 4.17 can be written mc ¼ h=l

17

(4.18)

Werner Heisenberg, born W€ urzburg, Germany, 1901. Ph.D. Munich, 1923. Professor, Leipzig University, Max Planck Institute. Nobel Prize 1932 for his famous uncertainty principle of 1927. Director of the German atomic bomb/reactor project 1939–1945. Held various scientific administrative positions in postwar (Western) Germany 1945–1970. Died Munich 1976. 18 Louis de Broglie, born Dieppe, 1892. Ph.D. University of Paris. Professor Sorbonne, Institut Henri Poincare´ (Paris). Nobel Prize in physics 1929. Died Paris, 1987.

4.2 The Development of Quantum Mechanics. The Schr€ odinger Equation

99

and because the product of mass and velocity is momentum, Eq. 4.18 can be written pp ¼ h=l

(4.19)

relating the momentum of a photon (in its particle aspect) to its wavelength (in its wave aspect). If Eq. 4.19 can be generalized to any particle, then we have p ¼ h=l



(4.20)

relating the momentum of a particle to its wavelength; this is the de Broglie equation. If a particle has wave properties it should be describable by somehow combining the de Broglie equation and a classical wave equation. A highly developed nineteenth century mathematical theory of waves was at Schr€odinger’s disposal, and the union of a classical wave equation with Eq. 4.20 was one of the ways that he derived his wave equation. Actually, it is said that the Schr€odinger equation cannot actually be derived, but is rather a postulate of quantum mechanics that can only be justified by the fact that it works [15]; this fine philosophical point will not be pursued here. Of his three approaches [15], Schr€ odinger’s simplest is outlined here. A standing wave (one with fixed ends like a vibrating string or a sound wave in a flute) whose amplitude varies with time and with the distance from the ends is described by d2 f ðxÞ 4p2 ¼  f ðxÞ dx2 l2

(4.21)

f(x) ¼ amplitude of the wave, x ¼ distance from some chosen origin, l ¼ wavelength From Eq. 4.20: l ¼ h=mv

(4.22)

l ¼ wavelength of particle of mass m and velocity v Identifying the wave with a particle and substituting for l from Eq. 4.22 into Eq. 4.21: d 2 f ðxÞ 4p2 m2 v 2 ¼ f ðxÞ 2 dx h2

(4.23)

Since the total energy of the particle is the sum of its kinetic and potential energies: Ekin ¼ E  Epot ¼ E  V

(4.24)

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4 Introduction to Quantum Mechanics in Computational Chemistry

E ¼ total energy of the particle and V ¼ potential energy (the usual symbol), i.e. 1 2 mv ¼ E  V 2

(4.25)

Substituting Eq. 4.25 for mv 2 into Eq. 4.23: d2 f ðxÞ 8p2 m ¼  2 ðE  V Þf ðxÞ 2 dx h

(4.26)

f(x) ¼ amplitude of the particle/wave at a distance x from some chosen origin, m ¼ mass of the particle, E ¼ total energy (kinetic þ potential) of the particle, V ¼ potential energy of the particle (possibly a function of x). This is the Schr€ odinger equation for one-dimensional motion along the spatial coordinate x. It is usually written d 2 c 8p2 m þ 2 ðE  V Þ c ¼ 0 dx2 h

(4.27)

c ¼ amplitude of the particle/wave at a distance x from some chosen origin The one-dimensional Schr€ odinger equation is easily elevated to three-dimensional status by replacing the one-dimensional operator d2/dx2 by its three-dimensional analogue @2 @2 @2 þ 2 þ 2 ¼ r2 2 @x @y @z

(4.28)

r2 is the Laplacian operator “del squared.” Replacing d2/dx2 by r2, Eq. 4.27 becomes r2 c þ

8p2 m ðE  V Þc ¼ 0 h2

(4:29)

This is a common way of writing the Schr€odinger equation. It relates the amplitude c of the particle/wave to the mass m of the particle, its total energy E and its potential energy V. The meaning of c itself is, arguably, unknown [2] but the currently popular interpretation of c2, due to Born (Section 2.3) and Pauli19 is that it is proportional to the probability of finding the particle near a point P(x, y, z) (recall that c is a function of x, y, z): Prob ðdx; dy; dzÞ ¼ c2 dx dy dz Z c2 dx dy dz Prob ðVÞ ¼

(4:30) (4.31)

V

19

Wolfgang Pauli, born Vienna, 1900. Ph.D. Munich 1921. Professor Hamburg, Zurich, Princeton, Zurich, . Best known for the Pauli exclusion principle. Nobel Prize 1945.Died Zurich 1958.

4.2 The Development of Quantum Mechanics. The Schr€ odinger Equation

101

The probability of finding the particle in an infinitesimal cube of sides dx, dy, dz is c2dxdydz, and the probability of finding the particle somewhere in a volume V is the integral over that volume of c2 with respect to dx, dy, dz (a triple integral); c2 is thus a probability density function, with units of probability per unit volume. Born’s interpretation was in terms of the probability of a particular state, Pauli’s the chemist’s usual view, that of a particular location. The Schr€ odinger equation overcame the limitations of the Bohr approach (see the beginning of Section 4.2.6): the quantum numbers follow naturally from it (actually the spin quantum number ms requires a relativistic form of the Schr€odinger equation, the Dirac equation, and electron “spin” is apparently not really due to the particle spinning like a top). The Schr€ odinger equation can be solved in an exact analytical way only for one-electron systems like the hydrogen atom, the helium monocation and the hydrogen molecule ion, but the mathematical approach is complicated and of no great relevance to the application of this equation to the study of serious molecules. However a brief account of the results for hydrogenlike atoms is in order. The standard approach to solving the Schr€ odinger equation for hydrogenlike atoms involves transforming it from Cartesian (x, y, z) to polar coordinates (r, y, ’), since these accord more naturally with the spherical symmetry of the system. This makes it possible to separate the equation into three simpler equations, f(r) ¼ 0, f(y) ¼ 0, and f(’) ¼ 0. Solution of the f(r) equation gives rise to the n quantum number, solution of the f(y) equation to the l quantum number, and solution of the f(’) equation to the mm (often simply called m) quantum number. For each specific n ¼ n0 , l ¼ l0 and mm ¼ mm0 there is a mathematical function obtained by combining the appropriate f(r), f(y) and f(’): cðr; y; ’; n0 ; I 0 ; m0m Þ ¼ f ðrÞf ðyÞf ð’Þ

(4.32)

The function c(r, y, ’) (clearly c could also be expressed in Cartesians), depends functionally on r, y, ’ and parametrically on n, l and mm: for each particular set (n0 , l0 , mm0 ) of these numbers there is a particular function with the spatial coordinates variables r, y, ’ (or x, y, z). A function like ksinx is a function of x and depends only parametrically on k. This c function is an orbital (“quasi-orbit”; the term was invented by Mulliken, Section 4.3.4), and you are doubtless familiar with plots of its variation with the spatial coordinates. Plots of the variation of c2 with spatial coordinates indicate variation of the electron density (recall the Born interpretation of the wavefunction) in space due to an electron with quantum numbers n0 , l0 and mm0 . We can think of an orbital as a region of space occupied by an electron with a particular set of quantum numbers, or as a mathematical function c describing the energy and the shape of the spatial domain of an electron. For an atom or molecule with more than one electron, the assignment of electrons to orbitals is an (albeit very useful) approximation, since orbitals follow from solution of the Schr€ odinger equation for a hydrogen atom. The Schr€ odinger equation that we have been talking about is actually the timeindependent (and nonrelativistic) Schr€ odinger equation: the variables in the equation are spatial coordinates, or spatial and spin coordinates (Section 5.2.3.1) when

102

4 Introduction to Quantum Mechanics in Computational Chemistry

electron spin is taken into account. The time-independent equation is the one most widely-used in computational chemistry, but the more general time-dependent Schr€ odinger equation, which we shall not examine, is important in certain applications, like some treatments of the interaction of a molecule with light, since light (radiation) is composed of time-varying electric and magnetic fields. The timedependent density functional theory method of calculating UV spectra (Chapter 7) is based on the time-dependent Schr€ odinger equation.

4.3

4.3.1

The Application of the Schr€ odinger Equation to Chemistry by H€ uckel Introduction

The quantum mechanical methods described in this book are all molecular orbital (MO) methods, or oriented toward the molecular orbital approach: ab initio and semiempirical methods use the MO method, and density functional methods are oriented toward the MO approach. There is another approach to applying the Schr€ odinger equation to chemistry, namely the valence bond method. Basically the MO method allows atomic orbitals to interact to create the molecular orbitals of a molecule, and does not focus on individual bonds as shown in conventional structural formulas. The VB method, on the other hand, takes the molecule, mathematically, as a sum (linear combination) of structures each of which corresponds to a structural formula with a certain pairing of electrons [16]. The MO method explains in a relatively simple way phenomena that can be understood only with difficulty using the VB method, like the triplet nature of dioxygen or the fact that benzene is aromatic but cyclobutadiene is not [17]. With the application of computers to quantum chemistry the MO method almost eclipsed the VB approach, but the latter has in recent years made a limited comeback [18]. The first application of quantitative quantum theory to chemical species significantly more complex than the hydrogen atom was the work of H€uckel20 on unsaturated organic compounds, in 1930–1937 [19]. This approach, in its simplest form, focuses on the p electrons of double bonds, aromatic rings and heteroatoms. Although H€ uckel did not initially explicitly consider orbital hybridization (the concept is usually credited to Pauling,21 1931 [20]), the method as it became widely applied [21] confines itself to planar arrays of sp2-hybridized atoms, usually carbon atoms, and evaluates the consequences of the interactions among the p electrons (Fig. 4.4). Actually, the simple H€ uckel method has been occasionally applied to 20

Erich H€uckel, born Berlin, 1896. Ph.D. G€ ottingen. Professor, Marburg. Died Marburg, 1980. Linus Pauling, born Portland, Oregon, 1901. Ph.D. Caltech. Professor, Caltech. Known for work in quantum chemistry and biochemistry, campaign for nuclear disarmament, and controversial views on vitamin C. Nobel Prize for chemistry 1954, for peace 1963. Died near Big Sur, California, 1994.

21

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel Fig. 4.4 The simple H€ uckel method is used mainly for planar arrays of p systems

103

i.e. C4H6

i.e. .

.

C4H6

i.e.

C6H6

z

1 2 i.e.

1 2

+

–

2s –

1 2

2s + (–

z

2py 1 2

) 2py –

z

2s +

1 2

+

y

2s +

1 2

2py

+ – y

y 2py

an sp hybrid

an sp hybrid

Fig. 4.5 Hybridization is forming new atomic orbitals, on an atom, by mathematically mixing (combining) “original” atomic orbitals on that atom. Mixing two orbitals gives two hybrid orbitals, and in general n AOs give n hybrid AOs. Orbitals are mathematical functions and so can be added and subtracted as shown in the figure

nonplanar systems [22]. Because of the importance of the concept of hybridization in the simple H€ uckel method a brief discussion of this concept is warranted.

4.3.2

Hybridization

Hybridization is the mixing of orbitals on an atom to produce new, “hybridized” (in the spirit of the biological use of the term), atomic orbitals. This is done mathematically but can be appreciated pictorially (Fig. 4.5). One way to justify the procedure theoretically is to recognize that atomic orbitals are vectors in the generalized

104

4 Introduction to Quantum Mechanics in Computational Chemistry

mathematical sense of being elements of a vector space [23] (if not in the restricted sense of the physicist as physical entities with magnitude and direction); it is therefore permissible to take linear combinations of these vectors to produce new members of the vector space. A good, brief introduction to hybridization is given by Streitwieser [24]. In a familiar example, a 2s orbital can be mixed with three 2p orbitals to give four hybrid orbitals; this can be done in an infinite number of ways, such as (from now on f will be used for atomic orbitals and c for molecular orbitals): ðs þ px þ py þ pz Þ ðs þ px þ py  pz Þ ðs þ px  py  pz Þ ðs þ px  py þ pz Þ

(4.33)

fa ¼ 12 ðs þ px þ 21=2 pz Þ fb ¼ 12 ðs þ px  21=2 pz Þ fc ¼ 12 ðs  px þ 21=2 py Þ fd ¼ 12 ðs  px  21=2 py Þ

(4.34)

f1 f2 f3 f4

¼ 12 ¼ 12 ¼ 12 ¼ 12

or

Both the set (4.33) and the set (4.34) consist of four sp3 orbitals, since the electron density contributions from the component s and p orbitals to the hybrid is, in each case (considering the squares of the coefficients; recall the Born interpretation of the square of a wavefunction, Section 4.2.6) in the ratio 1:3, i.e. 1/4 : 3(1/4) and 1/4 : (1/4 + 2/4), and in each set we have used a total of one s orbital, and one each of the px, py and pz orbitals. The total electron density from each component orbital is unity, e.g. for s, 4(1/4). Hybridization is purely a mathematical procedure, originally invented to reconcile the quantum mechanical picture of electron density in s, p, etc. orbitals with traditional views of directed valence. For example, it is sometimes said that in the absence of hybridization combining a carbon atom with four unpaired electrons with four hydrogen atoms would give a methane molecule with three equivalent, mutually perpendicular bonds and a fourth, different, bond (Fig. 4.6). Actually, this is incorrect: the 2s and three 2p orbitals of an unhybridized carbon along with the four 1s orbitals of four hydrogen atoms provide, without invoking hybridization, a tetrahedrally symmetrical valence electron distribution that leads to tetrahedral methane with four equivalent bonds (Fig. 4.6). In fact, it has been said “It is sometimes convenient to combine aos [atomic orbitals] to form hybrid orbitals that have well defined directional character and then to form mos [molecular orbitals] by combining these hybrid orbitals. This recombination of aos to form hybrids is never necessary ...” [25]. Interestingly, the MOs accommodating the four highest-energy electron pairs of methane (the eight valence

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

105

z 2pz 2s

x 2px y 2py

+ 4 H atoms (no hybridization)

H

H H H

C H

C H

H

H

WRONG

RIGHT

One C-H bond somehow skewed, and different in length and strength from the others, which are mutually perpendicular, like the p orbitals

The four C-H bonds tetrahedrally oriented and of equal length and strength

Fig. 4.6 Hybridization is not needed to explain bonding, e.g. the tetrahedral geometry of methane

electrons) are not equivalent in energy (not degenerate). This is an experimental fact that can be shown by photoelectron spectroscopy [26]. Instead of four orbitals of the same energy we have three degenerate orbitals and one lower in energy (and of course the almost undisturbed 1s core orbital of carbon). This surprising arrangement is a consequence of the fact that symmetry requires one combination (i.e. one MO) of carbon and hydrogen orbitals (essentially a weighted sum of the C2s and the four H1s orbitals) to be unique and the other three AO combinations (the other three MOs) to be degenerate (they involve the C2p and the H1s orbitals) [26, 27]. It must be emphasized that although the methane valence orbitals are energetically different, the electron and nuclear distribution is tetrahedrally symmetrical – the molecule indeed has Td (Section 2.6) symmetry. The four MOs formed directly from AOs are the canonical MOs. They are delocalized (spread out over the molecule), and do not correspond to the familiar four bonding Csp3/H1s MOs, each of which is localized between the carbon nucleus and a hydrogen nucleus. However, the canonical MOs can be mathematically manipulated to give the familiar localized MOs (Section 5.2.3.1).

106

4 Introduction to Quantum Mechanics in Computational Chemistry

Another example illustrates a situation somewhat similar to that we saw with methane, and what was until some years ago a serious controversy: the best way to represent the carbon/carbon double bond [28]. The currently popular way to conceptualize the C¼C bond has it resulting from the union of two sp2-hybridized carbon atoms (Fig. 4.7); the sp2 orbitals on each carbon overlap end-on forming a s bond and the p orbitals on each carbon overlap sideways forming a p bond. Note that the usual depiction of a carbon p orbital is unrealistically spindle-shaped, necessitating depicting overlap with connecting lines as in Fig. 4.7. Figure 4.8 shows a picture in better accord with the calculated electron density in the p orbital, i.e. corresponding to the square of the wavefunction. The two leftover sp2 orbitals can be used to bond to, say, hydrogen atoms, as shown. From this viewpoint the double bond is thus composed of a s bond and a p bond. However, this is not the only way to represent the C¼C bond. One can, for example, mathematically construct a carbon atom with two sp2 orbitals and two sp5 orbitals; the union of two such carbons gives a double bond formed from two sp5/sp5 bonds (Fig. 4.9), rather than from a s bond and a p bond. Which is right? They are only different ways of viewing the same thing: the electron density in the C¼C bond decreases smoothly from the central C/C axis in both models (Fig. 4.10), and the experimental 13 C/H NMR coupling constant for the C-H bond would, in both models, be predicted to correspond to about 33% s character in the orbital used by carbon to bond to hydrogen [29]. The ability of the hybridization concept to correlate and Cp / Cp overlap, equal above and below the C2H4 plane (the pi bond) Csp2 / H1s overlap

H

H C

H

and

H

H

C

C

C

H H

H Csp2 / H1s overlap

Csp2 / Csp2 overlap (the sigma bond)

H i.e.

H C

H

C H

Fig. 4.7 The currently popular view of the C/C double bond: an sp2/sp2 s bond and a p/p p bond. Compare this with Figs. 4.8 and 4.9

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

107 z

The C 2p electron density (the square of the wavefunction) looks more like this:

than like this: C

C

+

The wavefunction itself looks like

–

Hence p / p overlap looks like this:

C

C

x

rather than like this:

C

C

Fig. 4.8 The electron density is represented by the square of the mathematical function we call the orbital. A carbon 2p orbital is actually more buxom than its conventional representation, and two 2p orbitals overlap better than the usual picture, e.g. Fig. 4.7, suggests

sp 5

Csp 2 / H1s overlap

one sp 5 / sp 5 bond

H

H C

H Csp

2 / H1s

H

C

and

H C

H

C

H

H

overlap sp 5

f (sp 5) =

f (sp 2) =

1 6

s+

1 s+ 3

another sp 5 / sp 5 bond two banana bonds 5 6

p

2 p 3

Fig. 4.9 The C/C double bond can be built from two sp5 orbitals. The result is the same as using a s bond and a p bond (Fig. 4.7): see Fig. 4.10

rationalize acidities of hydrocarbons in terms of the s character of the carbon orbital in a C–H bond [29] is an example of the usefulness of this idea. Most of the systems studied by the simple H€ uckel method are essentially flat, as expected for sp2 arrays, and many properties of these molecules can be at least qualitatively understood by considering the in-plane s electrons of the overlapping sp2 orbitals to simply represent a framework that holds the perpendicular p orbitals, in which we are interested, in an orientation allowing neighboring p orbitals to overlap.

108

4 Introduction to Quantum Mechanics in Computational Chemistry pi bond

sp 5 / sp 5 bond

sigma bond C

C + C

C

C

C

+

C

C

sp 5 / sp 5 bond

Approximately

C

C

But more like C

C

Fig. 4.10 The model of a C/C double bond as a s/p bond is at bottom really equivalent to the sp5/sp5 þ sp5/sp5 model: both result in the same electron distribution, which is the physically significant thing. There are no gaps in electron density between the carbons: as the contribution to density from the s bond (or one of the sp5/sp5 bonds) falls off, the contribution from the p bond (or the other sp5/sp5 bond) increases. The electron density falls off smoothly with distance from the C/C axis. For some purposes one of the models, s/p or bent (banana) bonds, may be more useful

Before moving on to H€ uckel theory we take a look at matrices, since matrix algebra is the simplest and most elegant way to handle the linear equations that arise when MO theory is applied to chemistry.

4.3.3

Matrices and Determinants

Matrix algebra was invented by Cayley22 as a systematic way of dealing with systems of linear equations. The single equation in one unknown ax ¼ b has the solution 22

x ¼ a1 b

Arthur Cayley, lawyer and mathematician, born Richmond, England, 1821. Graduated Cambridge. Professor, Cambridge. After Euler and Gauss, history’s most prolific author of mathematical papers. Died Cambridge, 1895.

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

109

Consider next the system of two equations in two unknowns 1.

a11 x þ a12 y ¼ c1

2.

a21 x þ a22 y ¼ c2

The subscripts of the unknowns coefficients a indicate row 1, column 1, row 1, column 2, etc. We’ll see that using matrices the solutions (the values of x and y) can be expressed in a way analogous to that for the equation ax ¼ b. A matrix is a rectangular array of “elements” (numbers, derivatives, etc.) that obeys certain rules of addition, subtraction, and multiplication. Curved or angular brackets are used to denote a matrix: 0 1 5
 1 2 B C ð0 0 7 4Þ @2A 7 2 0 2  2 matrix

3  1 matrix

1  4 matrix

or: 

1 7

2 2



2 3 5 425 0

½0

0

7

4

Do not confuse matrices with determinants (below), which are written with straight lines, e.g.   1 2   7 3 is a determinant, not a matrix. This determinant represents the number 1  3  2  7 ¼ 3  14 ¼ 11. In contrast to a determinant, a matrix is not a number, but rather an operator (although some would consider matrices to be generalizations of numbers, with e.g. the 1  1 matrix (3) ¼ 3). An operator acts on a function (or a vector) to give a new function, e.g. d/dx acts on (differentiates) f(x) to give f 0 (x): d df ðxÞ f ðxÞ ¼ ¼ f 0 ðxÞ dx dx and the square root operator acts on y2 to give y. When we have done matrix multiplication you will see that a matrix can act on a vector and rotate it through an angle to give a new vector. Let’s look at matrix addition, subtraction, multiplication by scalars, and matrix multiplication (multiplication of a matrix by a matrix).

110

4.3.3.1

4 Introduction to Quantum Mechanics in Computational Chemistry

Addition and Subtraction

Matrices of the same size (2  2 and 2  2, 3  1 and 3  1, etc.) can be added just by adding corresponding elements:


2

1

7

4




þ



 2þ1 1þ3 3 4 ¼ ¼ 5 6 7þ5 4þ6 12 10 0 1 0 1 0 1 0 1 7 4 7þ4 11 B C B C B C B C @0A þ @4A ¼ @0 þ 4A ¼ @ 4 A 1

3



3

3þ1

1

4

Subtraction is similar:


4.3.3.2

2 7

1 4






1 5

3 6




¼

21 13 75 46




¼

2 2

1 2



Multiplication by a Scalar

pffi A scalar is an ordinary number (in contrast to a vector or an operator), e.g. 1, 2, 2, 1.714, p, etc. To multiply a matrix by a scalar we just multiply every element by the number:
2

4.3.3.3

2 7

1 4




¼

22 27

21 24




¼

4 14

2 8



Matrix Multiplication

We could define matrix multiplication to be analogous to addition: simply multiplying corresponding elements. After all, in mathematics any rules are permitted, as long as they do not lead to contradictions. However, as we shall see in a moment, for matrices to be useful in dealing with simultaneous equations we must adopt a slightly more complex multiplication rule. The easiest way to understand matrix multiplication is to first define series multiplication. If series a ¼ Sa ¼ a1 a2 a3 . . . , and series b ¼ Sb ¼ b1 b2 b3 . . . then we define the series product as Sa Sb ¼ a1 b1 þ a2 b2 þ a3 b3 þ :::

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

111

So for example, if Sa ¼ 5 2 1 and Sb ¼ 3 6 2 then Sa Sb ¼ 5(3) þ 2(6) þ 1(2) ¼ 15 þ 12 þ 2 ¼ 29 Now it’s easy to understand matrix multiplication: if AB ¼ C, where A, B, and C are matrices, then element i, j of the product matrix C is the series product of row i of A and column j of B. For example
AB ¼

1 7

3 2




2 5

4 6




¼

1ð2Þ þ 3ð5Þ 7ð2Þ þ 2ð5Þ

1ð4Þ þ 3ð6Þ 7ð4Þ þ 2ð6Þ




¼

17 24

22 40



(With practice, you can multiply simple matrices in your head.) Note that matrix multiplication is not commutative: AB is not necessarily BA, e.g.
BA ¼

2 5

4 6




1 7

3 2




¼

2ð1Þ þ 4ð7Þ 5ð1Þ þ 6ð7Þ

2ð3Þ þ 4ð2Þ 5ð3Þ þ 6ð2Þ




¼

30 47

14 27



(two matrices are identical if and only if their corresponding elements are the same). Note that two matrices may be multiplied together only if the number of columns of the first equals the number of rows of the second. Thus we can multiply A(2  2)B(2  2), A(2  2)B(2  3), A(3  1)B(1  3), and so on. A useful mnemonic is (a  b)(b  c) ¼ (a  c), meaning, for example that A(2  1) times B(1  2) gives C(2  2):

5 5ð0Þ ð0 3Þ ¼ 2 2ð0Þ

5ð3Þ 2ð3Þ




¼

0 0

15 6



It is helpful to know beforehand the size i.e. (2  2), (3  3), whatever, of the matrix you will get on multiplication. To get an idea of why matrices are useful in dealing with systems of linear equations, let’s go back to our system of equations 1.

a11 x þ a12 y ¼ c1

2.

a21 x þ a22 y ¼ c2

Provided certain conditions are met this can be solved for x and y, e.g. by solving (1) for x in terms of y then substituting for x in (2) etc. Now consider the equations from the matrix viewpoint. Since
AB ¼

a11 a21

a12 a22



x a11 x ¼ y a21 x

a12 y a22 y



clearly AB corresponds to the left hand side of the system, and the system can be written

112

4 Introduction to Quantum Mechanics in Computational Chemistry


AB ¼ C

C¼

where

c1 c2



A is the coefficients matrix, B is the unknowns matrix, and C is the constants matrix. Now, if we can find a matrix A1 such that A1 AB ¼ B (analogous to the numbers a1ab ¼ b) then A1 AB ¼ A1 C

i.e: B ¼ A1 C

Thus the unknowns matrix is simply the inverse of the coefficients matrix times the constants matrix. Note that we multiplied by A1 on the left (A1AB ¼ A1C), which is not the same as multiplying on the right, which would give ABA1 ¼ CA1; this is not necessarily the same as B. To see that a matrix can act as an operator consider the vector from the origin to the point P(3,4). This can be written as a column matrix, and multiplying it by the rotation matrix shown transforms it (rotates it) into another matrix: y vector

3 4

multiply on the left by rotation matrix

y

0 –1 1 0

x

4.3.3.4

new, rotated vector –4 3

x

Some Important Kinds of Matrices

These matrices are particularly important in computational chemistry: 1. 2. 3. 4. 5. 6. 7.

The zero matrix (the null matrix) Diagonal matrices The unit matrix (the identity matrix) The inverse of another matrix Symmetric matrices The transpose of another matrix Orthogonal matrices

1. The zero matrix or null matrix, 0, is any matrix with all its elements zero. Examples:


0 0 0 0






0 0

0 0

0 0

 ð0 0

0

0Þ

Clearly, multiplication by the zero matrix (when the (a  b)(b  c) mnemonic permits multiplication) gives a zero matrix.

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

113

2. A diagonal matrix is a square matrix that has all its off-diagonal elements zero; the (principal) diagonal runs from the upper left to the lower right. Examples: 0 1 0 1
 3 0 0 0 0 0 2 0 @0 6 0A @0 0 0A 0 4 0 0 1 0 0 0 3. The unit matrix or identity matrix 1 or I is a diagonal matrix whose diagonal elements are all unity. Examples: 0 1
 1 0 0 1 0 @0 1 0A ð1Þ 0 1 0 0 1 Since diagonal matrices are square, unit matrices must be square (but zero matrices can be any size). Clearly, multiplication (when permitted) by the unit matrix leaves the other matrix unchanged: 1A ¼ A1 ¼ A. 4. The inverse A1 of another matrix A is the matrix that, multiplied A, on the left or right, gives the unit matrix: A1A ¼ AA1 ¼ 1. Examples:

 2 1 1 2 If A ¼ then A1 ¼ 3=2 1=2 3 4 Check it out. 5. A symmetric matrix is a square matrix for which aij ¼ aji for each element. Examples: 0 1
 2 7 1 1 4 @7 3 5A a12 ¼ a21 ¼ 4 a12 ¼ a21 ¼ 7; etc: 4 3 1 5 4 Note that a symmetric matrix is unchanged by rotation about its principal diagonal. The complex-number analogue of a symmetric matrix is a Hermitian matrix (after the mathematician Charles Hermite); this has aij ¼ aji*, e.g. if element (2,3) ¼ a þ pffibi, then element (3,2) ¼ a  bi, the complex conjugate of element (2,3); i ¼ 1. Since all the matrices we will use are real rather than complex, attention has been focussed on real matrices here. ˜ ) of a matrix A is made by exchanging rows and columns. 6. The transpose (AT or A Examples:

 2 4 2 3 T If A ¼ then A ¼ 3 7 4 7 0 1
 2 1 2 1 6 If A ¼ then AT ¼ @ 1 7 A 1 7 2 6 2

114

4 Introduction to Quantum Mechanics in Computational Chemistry

Note that the transpose arises from twisting the matrix around to interchange rows and columns. Clearly the transpose of a symmetric matrix A is the same matrix A. For complex-number matrices, the analogue of the transpose is the conjugate transpose A{; to get this form A*, the complex conjugate of A, by converting each complex number element a þ bi in A to its complex conjugate a – bi, then switch the rows and columns of A* to get (A*)T ¼ A{. Physicists call A{ the adjoint of A, but mathematicians use adjoint to mean something else. 7. An orthogonal matrix is a square matrix whose inverse is its transpose: if A1 ¼ AT then A is orthogonal. Examples:
A1 ¼

0

p 1=p 6 A2 ¼ @ 2=p 6 1= 6

p p  1=p 2 1=p 2 1= 2 1= 2

p 1= 2 0p 1= 2

p 1 1=p 3 1= p3 A 1= 3

We saw that for the inverse of a matrix, A1A ¼ AA1 ¼ 1, so for an orthogonal matrix ATA ¼ AAT ¼ 1, since here the transpose is the inverse. Check this out for the matrices shown. The complex analogue of an orthogonal matrix is a unitary matrix; its inverse is its conjugate transpose. The columns of an orthogonal matrix are orthonormal vectors. This means that if we let each column represent a vector, then these vectors are mutually orthogonal and each one is normalized. Two or more vectors are orthogonal if they are mutually perpendicular (i.e. at right angles), and a vector is normalized if it is of unit length. Consider the matrix A1 above. If column 1 represents the vector v1 and column 2 the vector v2, then we can picture these vectors like this (the long side of a right triangle is of unit length if the squares of the other sides sum to 1): y 1 _ 2

_1 _ –1 2

v2

1 _ 2

v1

0

1 _ 2

1

v1 = x v2 =

1 _ 2 1 _ 2 _ –1 2 1 _ 2

The two vectors are orthogonal: from the diagram the angle between them is clearly 90 since the angle each makes with, say, the x-axis is 45 . Alternatively, the angle can be calculated from vector algebra: the dot product (scalar product) is v1  v2 ¼ jv1 kv2 j cos y where |v| (“mod v”) is the absolute value of the vector, i.e. its length: jvj ¼ ðv 2x þ v 2y Þ1=2 ðor ðv 2x þ v 2y þ v 2z Þ1=2 for a 3D vectorÞ: Each vector is normalized, i.e. jv1 j ¼ jv2 j ¼ ð12 þ 12Þ1=2 ¼ 1.

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

115

The dot product is also v1  v2 ¼ v 1x v 2x þ v 1y v 2y ðwith an obvious extension to 3D spaceÞ i.e. cos y ¼ ðv 1x v 2x þ v 1y v 2y Þ=jv1 kv2 j 



 1 1 1 1 p p ¼ p þ p =ð1Þð1Þ ¼ 0 2 2 2 2 and so y ¼ 90 Likewise, the three columns of the matrix A2 above represent three mutually perpendicular, normalized vectors in 3D space. A better name for an orthogonal matrix would be an orthonormal matrix. Orthogonal matrices are important in computational chemistry because molecular orbitals can be regarded as orthonormal vectors in a generalized n-dimensional space (Hilbert space, after the mathematician David Hilbert). We extract information about molecular orbitals from matrices with the aid of matrix diagonalization.

4.3.3.5

Matrix Diagonalization

Modern computer programs use matrix diagonalization to calculate the energies (eigenvalues) of molecular orbitals and the sets of coefficients (eigenvectors) that help define their size and shape. We met these terms, and matrix diagonalization, briefly in Section 2.5; “eigen” means suitable or appropriate, and we want solutions of the Schr€ odinger equation that are appropriate to our particular problem. If a matrix A can be written A ¼ PDP1, where D is a diagonal matrix (you could call P and P1 pre- and postmultiplying matrices), then we say that A is diagonalizable (can be diagonalized). The process of finding P and D (getting P1 from P is simple for the matrices of computational chemistry – see below) is matrix diagonalization. For example



 1 2 4 2 1 2 2 0 1 if A ¼ then P ¼ ; D¼ ; and P ¼ 1 2 1 1 1 1 0 3 Check it out. Linear algebra texts describe an analytical procedure using determinants, but computational chemistry employs a numerical iterative procedure called Jacobi matrix diagonalization, or some related method, in which the off-diagonal elements are made to approach zero. Now, it can be proved that if and only if A is a symmetric matrix (or more generally, if we are using complex numbers, a Hermitian matrix – see symmetric matrices, above), then P is orthogonal (or more generally, unitary – see orthogonal matrices, above) and so the inverse P1 of the premultiplying matrix P is simply the

116

4 Introduction to Quantum Mechanics in Computational Chemistry

transpose of P, PT (or more generally, what computational chemists call the conjugate transpose A{ – see transpose, above). Thus
 0 1 if A ¼ then 1 0


 0:707 0:707 0:707 0:707 1 0 P¼ ;D¼ ; P1 ¼ 0:707 0:707 0:707 0:707 0 1 (In this simple example the transpose of P happens to pbe ffi identical with P). In the spirit of numerical methods 0.707 is used instead of 1/ 2. A matrix like A above, for which the premultiplying matrix P is orthogonal (and so for which P1 ¼ PT) is said to be orthogonally diagonalizable. The matrices we will use to get molecular orbital eigenvalues and eigenvectors are orthogonally diagonalizable. A matrix is orthogonally diagonalizable if and only if it is symmetric; this has been described as “one of the most amazing theorems in linear algebra” (see Roman S (1988) An introduction to linear algebra with applications. Harcourt Brace, Orlando, p 408) because the concept of orthogonal diagonalizability is not simple, but that of a symmetric matrix is very simple.

4.3.3.6

Determinants

A determinant is a square array of elements that is a shorthand way of writing a sum of products; if the elements are numbers, then the determinant is a number. Examples:  a  11   a21

 a12   ¼ a11 a22  a12 a21 ; a22 

 5   4

 2   ¼ 5ð3Þ  2ð4Þ ¼ 7 3

As shown here, a 2  2 determinant can be expanded to show the sum it represents by “cross multiplication”. A higher-order determinant can be expanded by reducing it to smaller determinants until we reach 2  2 determinants; this is done like this:         2 1 3 0  1 7 5  1 3 5  7 3 5          1 7 3 5           ¼ 2 4 6 1   1 3 6 1  þ 3 3 4 1         3 4 6 1   1 8 2   1 2 2   8 2 2     1 8 2 2    1 7 3      0 3 4 6    1 8 2 Here we started with element (1,1) and moved across the first row. The first of the above four terms is 2 times the determinant formed by striking out the row and

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

117

column in which 2 lies, the second term is minus one times the determinant formed by striking out the row and column in which 1 lies, the third term is plus 3 times the determinant formed by striking out the row and column in which 3 lies, and the fourth term is minus 0 times the determinant formed by striking out the row and column in which 0 lies; thus starting with the element of row 1, column 1, we move along the row and multiply by þ1, 1, þ1, 1. It is also possible to start at, say element (2,1), the number 1, and move across the second row (, þ, , þ), or to start at element (1,2) and go down the column (, þ, , þ), etc. One would likely choose to work along a row or column with the most zeroes. The (n  1)  (n  1) determinants formed in expanding an n  n determinant are called minors, and a minor with its appropriate þ or  sign is a cofactor. Expansion of determinants using minors/cofactors is called Lagrange expansion (Joseph Louis Lagrange 1773). There are also other approaches to expanding determinants, such as manipulating them to make all the elements but one of a row or column zero; see any text on matrices and determinants. The third-order determinants in the example above can be reduced to second-order ones and so the fourth-order determinant can be evaluated as a single number. Obviously every determinant has a corresponding square matrix and every square matrix has a corresponding determinant, but a determinant is not a matrix; it is a function of a matrix, a rule that tells us how to take the set of numbers in a matrix and get a new number. Approaches to the study of determinants were made by Seki in Japan and Leibnitz in Europe, both in 1683. The word “determinant” was first used in our sense by Cauchy (1812), who also wrote the first definitive treatment of the topic.

4.3.3.7

Some Properties of Determinants

These are stated in terms of rows, but also hold for columns; D is “the determinant”. 1. If each element of a row is zero, D is zero (obvious from Lagrange expansion). 2. Multiplying each element of a row by k multiplies D by k (obvious from Lagrange expansion). 3. Switching two rows changes the sign of D (since this changes the sign of each term in the expansion). 4. If two rows are the same D is zero. (follows from 3, since if n ¼ n, n must be zero. 5. If the elements of one row are a multiple of those of another, D is zero (follows from 2 and 4). 6. Multiplying a row by k and adding it (adding corresponding elements) to another row causes no further change in D (in the Laplace expansion the terms without k cancel). 7. A determinant A can be written as the sum of two determinants B and C which differ only in row i in accordance with this rule: if row i of A is bi1 þ ci1 bi2 þ ci2 . . . then row i of B is bi1 bi2 . . . and row i of C is ci1 ci2 . . . An example makes this clear; with row i ¼ row 3:

118

4 Introduction to Quantum Mechanics in Computational Chemistry

 1  5  8

4.3.4

3 4 11

  6   1 2  ¼  5 9 5 þ 3

3 4 7þ4

  6   1 2  ¼  5 4 þ 5 5

3 4 7

  6   1 2  þ  5 4 3

3 4 4

 6  2  5

The Simple H€ uckel Method – Theory

The derivation of the H€ uckel method (SHM, or simple H€uckel theory, SHT; also called H€ uckel molecular orbital method, HMO method) given here is not rigorous and has been strongly criticized [30a]; nevertheless it has the advantage of showing how with simple arguments one can use the Schr€ odinger equation to develop, more by a plausibility argument than a proof, a method that gives useful results and which can be extended to more powerful methods with the retention of many useful concepts from the simple approach. The Schr€ odinger equation (Section 4.2.6, Eq. 4.29) r2 c þ

8p2 m ðE  VÞc ¼ 0 h2

can after very simple algebraic manipulation be rewritten
 h2 2  2 r þ V c ¼ Ec 8p m

(4.35)

This can be abbreviated to the seductively simple-looking form ^ ¼ Ec Hc

(4:36)

where H^ ¼




h2 r2 þ V 8p2 m

 (4:37)

The symbol Hˆ (“H hat” or “H peak”) is an operator (Section 4.3.3): it specifies that an operation is to be performed on c, and Eq. 4.36 says that the result of the operation will be E multiplied by c. The operation to be performed on c (i.e. c(x,y, z)) is “differentiate it twice with respect to x, to y and to z, add the partial derivatives, and multiply the sum by h2/8p2m; then add this result to V times c” (now you can see why symbols replaced words in mathematical discourse). The notation Hˆc means Hˆ of c, not Hˆ times c. Equation 4.36 says that an operator (Hˆ) acting on a function (c) equals a constant (E) times the function (“H hat of psi equals E psi”). Such an equation

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

^ ¼ kf ; Of

O^ ¼ operator

119

(4.38)

is called an eigenvalue equation. The functions f and constants k that satisfy ^ Eq. (4.38) are eigenfunctions and eigenvalues, respectively, of the operator O. ^ The operator H is called the Hamiltonian operator, or simply the Hamiltonian. The term is named after the mathematician Sir William Rowan Hamilton, who formulated Newton’s equations of motion in a manner analogous to the quantum mechanical equation 4.36. Eigenvalue equations are very important in quantum mechanics, and we shall again meet eigenfunctions and eigenvalues. The eigenvalue formulation of the Schr€ odinger equation is the starting point for our derivation of the H€ uckel method. We will apply Eq. 4.36 to molecules, so in this context Hˆ and c are the molecular Hamiltonian and wavefunction, respectively. From ^ ¼ Ec Hc we get ^ ¼ Ec2 cHc

(4.39)

Note that this is not the same as Hˆc2 ¼ Ec2, just as x df(x)/dx, say, is not the same as dxf(x)/dx. Integrating and rearranging we get R ^ cHcdv E¼ R 2 c dv

(4.40)

The integration variable dv indicates integration with respect to spatial coordinates (x, y, z in a Cartesian coordinate system), and integration over all of space is implied, since that is the domain of an electron in a molecule, and thus the domain of the variables of the function c. One might wonder why not simply use E ¼ Hˆc/c; the problems with this function are that it goes to infinity as c approaches zero, and it is not well-behaved with regard to finding a minimum by differentiation. Next we approximate the molecular wavefunction c as a linear combination of atomic orbitals (LCAO). The molecular orbital (MO) concept as a tool in interpreting electronic spectra was formalized by Mulliken23 starting in 1932 and building on earlier (1926) work by Hund24 [31] (recall that Mulliken coined the word 23

Robert Mulliken, born Newburyport, Massachusetts, 1896. Ph.D. University of Chicago. Professor New York University, University of Chicago, Florida State University. Nobel Prize in chemistry 1966, for the MO method. Died Arlington, Virginia, 1986. 24 Friedrich Hund, born Karlsruhe, Germany, 1896. Ph.D. Marburg, 1925, Professor Rostock, Leipzig, Jena, Frankfurt, G€ ottingen. Died G€ ottingen, 1997.

120

4 Introduction to Quantum Mechanics in Computational Chemistry

orbital). The postulate behind the LCAO approach is that an MO can be “synthesized” by combining simpler functions, now called basis functions; these functions comprise a basis set. This way of calculating MOs is based on suggestions of Pauling (1928) [32] and Lennard–Jones25 (1929) [33]. Perhaps the most important early applications of the LCAO method were the simple H€uckel method (1931) [19], in which p AOs orbitals are combined to give p AOs (probably the first time that the MOs of relatively big molecules were represented as a weighted sum of AOs with optimized coefficients), and the treatment of all the lower electronic states of the hydrogen molecule by Coulson26 and Fischer (1949) [34]. The basis functions are usually located on the atoms of the molecule, and may or may not (see the discussion of basis functions in Section 5.3) be conventional atomic orbitals. The wavefunction can in principle be approximated as accurately as desired by using enough suitable basis functions. In this simplified derivation of the H€uckel method we at first consider a molecule with just two atoms, with each atom contributing one basis function to the MO. Combining basis functions on different atoms to give MOs spread over the molecule is somewhat analogous to combining atomic orbitals on the same atom to give hybrid atomic orbitals (Section 4.3.2) [27]. The combination of n basis functions always gives n MOs, as indicated in Fig. 4.11, and we expect two MOs for the two-atomic-orbital diatomic molecule we are using here. Using the LCAO approximation c ¼ c1 f1 þ c2 f2

(4.41)

where f1 and f2 are basis functions on atoms 1 and 2, and c1 and c2 are weighting coefficients to be adjusted to get the best c, and substituting into Eq. 4.40 we get R E¼

^ 1 f1 þ c2 f2 Þdv ðc1 f1 þ c2 f2 ÞHðc R ðc1 f1 þ c2 f2 Þ2 dv

(4.42)

If we multiply out the terms in Eq. 4.42 we get E¼

25

c21 H11 þ 2c1 c2 H12 þ c22 H22 c21 S11 þ 2c1 c2 S12 þ c22 S22

(4.43)

John Edward Lennard-Jones, born Leigh, Lancaster, England, 1894. Ph.D. Cambridge, 1924. Professor Bristol. Best known for the Lennard–Jones potential function for nonbonded atoms. Died Stoke-on-Trent, England, 1954. 26 Charles A. Coulson, born Worcestershire, England, 1910. Ph.D. Cambridge, 1935. Professor of theoretical physics, King’s College, London; professor of mathematics, Oxford; professor of theoretical chemistry, Oxford. Died Oxford, 1974. Best known for his book "Valence" (the 1st Ed., 1952).

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

121

where Z Z Z Z Z Z

^ 1 dv ¼ H11 f1 Hf ^ 2 dv ¼ H12 ¼ f1 Hf

Z

^ 1 dv ¼ H21 f2 Hf

^ 2 dv ¼ H22 f2 Hf (4.44) f21 dv ¼ S11 f1 f2 dv ¼ S12 ¼

Z f2 f1 dv ¼ S21

f22 dv ¼ S22 Note that in Eqs. 4.43 and 4.44 the Hij are not operators hence are not given hats; they are integrals involving Hˆ and basis functions f. For any particular molecular geometry (i.e. nuclear configuration: Section 2.3, the Born–Oppenheimer approximation) the energy of the ground electronic state is the minimum energy possible for that particular nuclear arrangement and the collection of electrons that goes with it. Our objective now is to minimize the energy with respect to the basis set coefficients. We want to find the c’s corresponding to the minimum on an energy versus c’s potential energy surface. To do this we follow a standard calculus procedure: set ∂E/∂c1 equal to zero, explore the consequences, then repeat for ∂E/∂c2. In theory, setting the first derivatives equal to zero guarantees only that we will find in “MO space” (an abstract space defined by an energy axis and two or more coefficient axes) a stationary point (cf. Section 2.2), but examining the second derivatives shows that the procedure gives an energy minimum if all or most of the electrons are in bonding MOs, which is the case for most real molecules [35]. Write Eq. 4.43 as  E c21 S11 þ 2c1 c2 S12 þ c22 S22 ¼ c21 H11 þ 2c1 c2 H12 þ c22 H22

(4.45)

and differentiate with respect to c1:
 @E  2 c1 S11 þ 2c1 c2 S12 þ c22 S22 þ Eð2c1 S11 þ 2c2 S22 Þ ¼ 2c1 H11 þ 2c2 H12 @c1 Set @E=@c1 ¼ 0: Eð2c1 S11 þ 2c2 S22 Þ ¼ 2c1 H11 þ 2c2 H12 This can be written ðH11  ES11 Þc1 þ ðH12  ES12 Þc2 ¼ 0

(4.46)

122

4 Introduction to Quantum Mechanics in Computational Chemistry energy

Two s AOs AO 1

AO 2

+

+

f1

f2

two sigma-type MOs C12 f1 + C22f2

+

–

antibonding MO y2 MO 2

a node (AOs change sign here) C11f1 + C21f2

coefficient of basis function 1 in MO 1

+

+

coefficient of basis function 2 in MO 1

bonding MO y1 MO 1

two pi-type MOs Two p AOs AO 1

AO 2

+

+

– f1

f2

–

+ C12f2 + C22f2

C11f1 + C21f2

–

energy

–

node

+

+

+

–

–

antibonding MO y2 MO 2

bonding MO y1 MO 1

three pi-type MOs

Three p AOs AO 1

AO 2

AO 3

+

+

+

–

–

–

f1

f2

f3

+

–

C13f1 + C23f2 + C33f3

–

+

C12f1 + C22f2 + C32f3

+ –

nodes C22 = 0

energy

+

antibonding MO y3 – MO 3 –

+

nonbonding MO y2 MO 2

C11f1 + C21f2 + C31f3 + –

+

+

–

–

bonding MO y1 MO 1

Fig. 4.11 Linear combination of n atomic orbitals (or, more generally, basis functions) gives n MOs. The coefficients c are weighting factors that determine the magnitude and the sign of the contribution from each basis function. The functions contributing to the MO change sign at a node (actually a nodal plane) and the energy of the MOs increases with the number of nodes

The analogous procedure, beginning with Eq. 4.45 and differentiating with respect to c2 leads to ðH12  ES12 Þc1 þ ðH22  ES22 Þc2 ¼ 0

(4.47)

Equation 4.47 can be written as Eq. 4.48 ðH21  ES21 Þc1 þ ðH22  ES22 Þc2 ¼ 0

(4.48)

since as shown in Eqs. 4.44 H12 ¼ H21 and S12 ¼ S21, and the form used in Eq. 4.48 is preferable because it makes it easy to remember the pattern for the two-basis

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

123

function system examined here and for the generalization (see below) to n basis functions. Equations 4.46 and 4.48 form a system of simultaneous linear equations: ðH11  ES11 Þc1 þ ðH12  ES12 Þc2 ¼ 0 (4.49) ðH21  ES21 Þc1 þ ðH22  ES22 Þc2 ¼ 0 The pattern is that the subscripts correspond to the row and column in which they lie; this is literally true for the matrices and determinants we will consider later, but even for the system of equations 4.49 we note that in the first equation (“row 1”), the coefficient of c1 has the subscripts 11 (row 1, column 1) and the coefficient of c2 has the subscripts 12 (row 1, column 2), while in the second equation (“row 2”) the coefficient of c1 has the subscripts 21 (row 2, column 1) and the coefficient of c2 has the subscripts 22 (row 2, column 2). The system of equations 4.49 are called secular equations, because of a supposed resemblance to certain equations in astronomy that treat the long-term motion of the planets; from the Latin saeculum, a long period of time (not to be confused with secular meaning worldly as opposed to religious, which is from the Latin secularis, worldly, temporal). From the secular equations we can find the basis function coefficients c1 and c2, and thus the MOs c, since the c’s and the basis functions f make up the MOs (Eq. 4.41). The simplest, most elegant and most powerful way to get the coefficients and energies of the MOs from the secular equations is to use matrix algebra (Section 4.3.3). The following exposition may seem a little involved, but it must be emphasized that in practice the matrix method is implemented automatically on a computer, to which it is highly suited. The secular equations 4.49 are equivalent to the single matrix equation


H11  ES11 H21  ES21

H12  ES12 H22  ES22




c1 c2




 0 ¼ 0

(4.50)

Since the H  ES matrix is an H matrix minus an ES matrix, and since the ES matrix is the product of an S matrix and the scalar E, Eq. 4.50 can be written: 


H11 H21

H12 H22






S11 S21



 S12 c 0 E 1 ¼ S22 c2 0

(4.51)

which can be more concisely rendered as ½H  SEc ¼ 0

(4.52)

Hc ¼ SEc

(4.53)

and Eq. 4.52 can be written

124

4 Introduction to Quantum Mechanics in Computational Chemistry

H and S are square matrices and c and 0 are column matrices (Eq. 4.51), and E is a scalar (an ordinary number). We have been developing these equations for a system of two basis functions, so there should be two MOs, each with its own energy and its own pair of c’s (Fig. 4.11). We need two energy values and four c’s: we want to be able to calculate c11 and c21 of c1 (MO1, energy level 1) and c12 and c22 of c2 (MO2, 0 energy level; in keeping with common practice the energies of the MOs are designated e1 and e2. Equation 4.53 can be extended (our simple derivation shortchanges us here) [36] to encompass the four c’s and two e’s; the result is HC ¼ SC«

(4:54)

We now have only square matrices; in Eq. 4.53 c was a column matrix and E was not a matrix, but rather a scalar – an ordinary number. The matrices are (four equations Eqs (4.55)):
H¼

H11

H21 c11 C¼ c21
S11 S¼ S21
e1 e¼ 0


H12



H22  c12 c22  S12 S22  0 e2

(4:55)

The H matrix is an energy-elements matrix, the Fock27 matrix, whose elements are integrals Hij (Eqs. 4.44). Fock actually pointed out the need to take electron spin into account in more elaborate calculations than the simple H€uckel method; we will meet “real” Fock matrices in Chapter 5. For now, we just note that in the simple (and extended) H€uckel methods as an ad hoc prescription at most two electrons, paired, are allowed in each MO. Each Hij represents some kind of energy term, since Hˆ is an energy operator (Section 4.3.3). The meaning of the Hij’s is discussed later in this section. The C matrix is the coefficient matrix, whose elements are the weighting factors cij that determine to what extent each basis function f (roughly, each atomic orbital on an atom) contributes to each MO c. Thus c11 is the coefficient of f1 in c1, c21 the coefficient of f2 in c1, etc., with the first subscript indicating the basis function and the second subscript the MO (Fig. 4.11). In each column of C the c’s belong to the same MO. 27

Vladimer Fock, born St. Petersburg, 1898. Ph.D. Petrograd University, 1934. Professor Leningrad University, also worked at various institutes in Moscow. Worked on quantum mechanics and relativity, e.g. the Klein–Fock equation for particles with spin in an electromagnetic field. Died Leningrad, 1974.

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

125

The S matrix is the overlap matrix, whose elements are overlap integrals Sij which are a measure of how well pairs of basis functions (roughly, atomic orbitals) overlap. Perfect overlap, between identical functions on the same atom, corresponds to Sii ¼ 1, while zero overlap, between different functions on the same atom or well-separated functions on different atoms, corresponds to Sij ¼ 0. The diagonal e matrix is an energy-levels matrix, whose diagonal elements are MO energy levels ei, corresponding to the MOs ci. Each ei is ideally the negative of the energy needed to remove an electron from that orbital, i.e. the negative of the ionization energy for that orbital. Thus it is ideally the energy of an electron attracted to the nuclei and repelled by the other electrons, relative to the energy of that electron and the corresponding ionized molecule, infinitely separated from one another. This is seen by the fact that photoelectron spectra correlate well with the energies of the occupied orbitals, in more elaborate (ab initio) calculations [26]. In simple H€ uckel calculations, however, the quantitative correlation is largely lost. Now suppose that the basis functions f had these properties (the H and S integrals, involving f, are defined in Eqs. 4.44): S11 ¼ 1 S12 ¼ S21 ¼ 0

(4.56)

S22 ¼ 1 More succinctly, suppose that Sij ¼ dij

(4.57)

where dij is the Kronecker delta (Leopold Kronecker, German mathematician, ca. 1860) which has the property of being 1 or 0 depending on whether i and j are the same or different. Then the S matrix (Eqs. 4.55) would be
S¼

1 0

0 1

 (4.58)

Since this is a unit matrix Eq. 4.54 would become HC ¼ Ce

(4.59)

and by multiplying on the right by the inverse of C we get H ¼ CeC1

(4:60)

So from the definition of matrix diagonalization, diagonalization of the H matrix will give the C and the « matrices, i.e. will give the coefficients c and the MO

126

4 Introduction to Quantum Mechanics in Computational Chemistry

+

H

H +

+ C1 –

+

C3

C

C –

H

C2

C

–

H

H

*

a twisted allyl species * = +, .or –

Fig. 4.12 We cannot simply choose a set of orthonormal basis functions, because in a typical molecule many pairs of basis functions will not be orthogonal, i.e. will not have zero overlap. In the allyl species shown, the 2s and the 2p functions (i.e. AOs) on C1 are orthogonal (the þ part of the p orbital cancels the  part in overlap with the s orbital; in general AOs on the same atom are orthogonal), and the 2p functions on C2 and C3 are also orthogonal, if their axes are at right angles. However, the C1(2s)/C2(2p) and the C1(2p)/C2(2p) pairs are not orthogonal

energies e (Eqs. 4.55), if Sij ¼ dij (Eq. 4.57). This is a big if, and in fact it is not true. Sij ¼ dij would mean that the basis functions are both orthogonal and normalized, i.e. orthonormal. Orthogonal atomic (or molecular) orbitals or functions f have R zero net overlap (Fig. 4.12), corresponding to f f dv ¼ 0. A normalized orbital i j R or function f has the property ffdv ¼ 1. We can indeed use a set of normalized basis functions: a suitable normalization constant k applied to an unnormalized basis function f0 will ensure normalization (f ¼ kf0 ). However, we cannot choose a set of orthogonal atom-centered basis functions, because orthogonality implies zero overlap between the two functions in question, and in a molecule the overlap between pairs of basis functions will depend on the geometry of the molecule (Fig. 4.12). (However, as we will see later, the basis functions can be manipulated mathematically to give combinations of the original functions which are orthonormal). The assumption of basis function orthonormality is a drastic approximation, but it greatly simplifies the H€ uckel method, and in the present context it enables us to reduce Eq. 4.54 to Eq. 4.59, and thus to obtain the coefficients and energy levels by diagonalizing the Fock matrix. Later we will see that in the absence of the orthogonality assumption the set of basis functions can be mathematically transformed so that a modified Fock matrix can be diagonalized; in the simple H€uckel method we are spared this transformation. In the matrix approach to the H€uckel method, then, we must diagonalize the Fock matrix H; to do this we have to assign numbers to the matrix elements Hij, and this brings us to other simplifying assumptions of the SHM, concerning the Hij. In the SHM the energy integrals Hij are approximated as just three quantities (the units are, e.g., kJ mol1):

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

a, the coulomb integral Z ^ i dv ¼ Hii ¼ a fi Hf

i:e: basis functions on the same atom;

127

(4:61a)

b, the bond integral or resonance integral Z Z ^ ^ i dv ¼ Hji ¼ b fi Hfj dv ¼ Hij ¼ fj Hf

(4:61b)

for basis functions on adjacent atoms; Z Z ^ ^ i dv ¼ Hji ¼ 0 fi Hfj dv ¼ Hij ¼ fj Hf

(4:61c)

for basis functions neither on the same or on adjacent atoms. To give these approximations some physical significance, we must realize that in quantum mechanical calculations the zero of energy is normally taken as corresponding to infinite separation of the particles of a system. In the simplest view, a, the coulomb integral, is the energy of the molecule relative to a zero of energy taken as the electron and basis function (i.e. AO; in the simple H€uckel method, f is usually a carbon p AO) at infinite separation. Since the energy of the system actually decreases as the electron falls from infinity into the orbital, a is negative (Fig. 4.13). The negative of a, in this view, is the ionization energy (a positive quantity) of the orbital (the ionization energy of the orbital is defined as the energy needed to remove an electron from the orbital to an infinite distance away). Energy

0 electron falls from infinite distance into a p AO on C

e–

electron falls from infinite distance into a MO formed by overlap of two p AOs on adjacent carbons

C C

C

a = Hii < 0 kJ mol–1 b = Hij < 0 kJ mol–1

Fig. 4.13 The coulomb integral a is most simply (but not too accurately) viewed as the energy of an electron in a carbon 2p orbital, relative to its energy an infinite distance away. The bond integral (resonance integral) b is most simply (but not too accurately) viewed as the energy of an electron in an MO formed by adjacent 2p orbitals, relative to its energy an infinite distance away

128

4 Introduction to Quantum Mechanics in Computational Chemistry

The quantity b, the bond integral or resonance integral, is, in the simplest view, the energy of an electron in the overlap region (roughly, a two-center MO) of adjacent p orbitals relative to a zero of energy taken as the electron and two-center MO at infinite separation. Like a, b is a negative energy quantity. A rough, naive estimate of the value of b would be the average of the ionization energies (a positive quantity) of the two adjacent AOs, multiplied by some fraction to allow for the fact that the two orbitals do not coincide but are actually separated. These views of a and b are oversimplifications [30]. We derived the 2  2 matrices of Eqs. 4.55 starting with a two-orbital system. These results can be generalized to n orbitals: 0 B B H¼B B @

H11

H12

. . . H1n

H21

H22 .. . Hn2

. . . H2n .. ... . . . . Hnn

.. . Hn1

1 C C C C A

(4.62)

The H elements of Eq. 4.62 become a, b, or 0 according to the rules of Eqs. 4.61. This will be clear from the examples in Fig. 4.14.

a H

H C

H

a b

2

1

C

b a

H

b

a b 0 2 * * = + or . or –

b

a b

0

b a

a

b 0

3

1

c 4

1

b a 0

b a b

b 0 3

b

b 0 b a

2

Fig. 4.14 Some conjugated molecules, their p orbital arrays, simplified representations of the molecules, and their simple H€ uckel Fock matrices. Same-atom interactions are a, adjacent-atom interactions are b, and all other interactions are 0. To diagonalize the matrices, we use a ¼ 0 and b ¼ 1

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

129

The computer algorithms for matrix diagonalization use some version of the Jacobi rotation method [37], which proceeds by successive numerical approximations (textbooks describe a diagonalization method based on expanding the determinant corresponding to the matrix; this is not used in computational chemistry). Therefore in order to diagonalize our Fock matrices we need numbers in place of a and b. In methods more advanced than the SHM, like the extended H€uckel method (EHM), other semiempirical methods, and ab initio methods, the Hij integrals are calculated to give numerical (in energy units) values. In the SHM we simply use energy values in |b| units relative to a (recall that b is a negative quantity: Fig. 4.13). The matrix of Fig. 4.14a then becomes
H¼

a b

b a




¼

1 0

0 1

 (4.63)

An electron in an MO represented by a 1,2-type interaction is lower in energy than one in a p orbital (1,1-type interaction) by one |b| energy unit. Similarly, the H matrix of Fig. 4.14b becomes 0

0 1

1

0

B C H ¼ @ 1 0 1 A 0 1 0

(4.64)

and the H matrix of Fig. 4.14c becomes 0

0 B 1 H¼B @ 0 1

1 0 0 1 1 0 0 1

1 1 0 C C 1 A 0

(4.65)

The H matrices can be written down simply by setting all i, i-type interactions equal to 0, and all i, j-type interactions equal to 1 where i and j refer to atoms that are bonded together, and equal to 0 when i and j refer to atoms that are not bonded together. Diagonalization of the two-basis-function matrix of Eq. 4.63 gives
H¼

0

1

1

0




¼

0:707

0:707

0:707

0:707 C




1

0

0

1 e




0:707

0:707

0:707

0:707



C1 (4.66)

Comparing Eq. 4.66 with Eq. 4.60, we see that we have obtained the matrices we want: the coefficients matrix C and the MO energy levels matrix «. The columns of

130

4 Introduction to Quantum Mechanics in Computational Chemistry

C are eigenvectors, and the diagonal elements of « are eigenvalues; cf. Eq. 4.38 and the associated discussion of eigenfunctions and eigenvalues. The result of Eq. (4.66) is readily checked by actually multiplying the matrices (multiplication here is aided by knowing that an analytical rather than numerical diagonalization pffi shows that 0.707 are approximations to 1/ 2). Note that CC1 ¼ 1, and that C1 is the transpose of C. The first eigenvector of C, the left-hand column, corresponds to the first eigenvalue of «, the top left element; the second eigenvector corresponds to the second eigenvalue. The individual eigenvectors, v1 and v2, are column matrices:

 0:707 0:707 1 and 1 0:707 0:707 (4.67) v2 v1 Figure 4.15 shows a common way of depicting the results for this two-orbital calculation. Since the coefficients are weighting factors for the contributions of the basis functions to the MOs (Fig. 4.11 and associated discussion), the c’s of eigenvector v1 combine with the basis functions to give MO1 (c1) and the c’s of eigenvector v2 combine with these same basis functions to give MO2 (c2). MOs below a are bonding and MOs above a are antibonding. The « matrix translates into an energy level diagram with c1 of energy a + b and c2 of energy a  b, i.e. the MOs lie one |b| unit below and one |b| above the nonbonding a level. Since b, like a, is negative, the a + b and a  b levels are of lower and higher energy, respectively, than the nonbonding a level. energy a-b

y 2 = 0.707 f1 – 0.707 f 2

+

–

e=1

C

C

–

+

antibonding MO

nonbonding level

a

a+b

y 2 = 0.707 f1 + 0.707 f 2

+

+

e = –1

C

C

–

–

bonding MO

Fig. 4.15 The p molecular orbitals and p energy levels for a two-p-orbital system in the simple H€uckel method. The MOs are composed of the basis functions (two p AOs) and the eigenvectors, while the energies of the MOs follow from the eigenvalues (Eq. 4.66). The paired arrows represent a pair of electrons of opposite spin (in the electronic ground state of the neutral ethene molecule c1 is occupied and c2 is empty)

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

131

Diagonalization of the three-basis function matrix of Eq. 4.64 gives 0

0 1 0

1

B C @ 1 0 1 A ¼ 0 1 0 0 10 0:500 0:707 0:500 1:414 0 B CB 0 0:707 A@ 0 0 @ 0:707

0

0

v1

e1; 0;

0; e2 ;

0 0

0;

0;

e3

v3

0:500 0:707

1

C1

e

C

0:500

CB C 0 A@ 0:707 0 0:707 A 0 1:414 0:500 0:707 0:500

0:500 0:707 0:500 v2

10

(4.68) The energy levels and MOs corresponding to these results are shown in Fig. 4.16. Diagonalization of the four-basis-function matrix of Eq. 4.65 gives energy

y 3 = 0.500 f 1 – 0.707 f 2 + 500 f 3 e = 1.414

antibonding MO

a -b

a

a +b

y 2 = 0.700 f 1 + 0.000 f 2 – 0.700 f 3 e=0

nonbonding MO

y 3 = 0.500 f 1 + 0.707 f 2 + 0.500 f 3 e = –1.414

bonding MO

+ C –

–

C +

+ C –

–

+

C –

C

C +

+ C –

+ C

+ C –

–

Fig. 4.16 The p molecular orbitals and p energy levels for an acyclic three-p-orbital system in the simple H€uckel method. The MOs are composed of the basis functions (three p AOs) and the eigenvectors (the c’s), while the energies of the MOs follow from the eigenvalues (Eq. 4.68). In the drawings of the MOs, the relative sizes of the AOs in each MO suggest the relative contribution of each AO to that MO. This diagram is for the propenyl radical. The paired arrows represent a pair of electrons of opposite spin, in the fully-occupied lowest MO, c1, and the single arrow represents an unpaired electron in the nonbonding MO, c2; the highest p MO, c3, is empty in the radical

132

4 Introduction to Quantum Mechanics in Computational Chemistry

0

0 1 0 1

1

B 1 0 1 0 C B C B C¼ @ 0 1 0 1 A 1 0 10 0:500 0:500 0:500 0:500 2 B 0:500 0:500 0:500 0:500 CB 0 B CB B CB @ 0:500 0:500 0:500 0:500 A@ 0 0

0:500 0:500 0:500 0:500

1 0 1 10 000 0:500 0:500 0:500 0:500 B C 0 0 0C CB 0:500 0:500 0:500 0:500 C C CB 0 0 0 A@ 0:500 0:500 0:500 0:500 A

0 002

0:500 0:500 0:500 0:500

e1 0 0 0 v1

v2

v3

0 e2 0 0

v4

0 0 e3 0 0 0 0 e4 C 1

e

C

ð4:69Þ

The energy levels and MOs from these results are shown in Fig. 4.17. Note that all these matrix diagonalizations yield orthonormal eigenvectors: vivi ¼ 1 and vivj ¼ 0, as required the fact that the Fock matrices are symmetric (see the discussion of matrix diagonalization in Section 4.3.3). energy

a - 2b

a-b

a

a+b

y 4 = 0.500 j1 – 0.500 j2 + 0.500 j3 – 0.500 j4 e=2

+ C –

antibonding MO

y 3 = 0.500 j1 + 0.500 j2 – 0.500 j3 – 0.500 j4 nonbonding MOs

e=0

e=0

y 2 = 0.500 j1 – 0.500 j2 – 0.500 j3 + 0.500 j4

+ C + – C –

y 1 = 0.500 j1 + 0.500 j2 + 0.500 j3 + 0.500 j4 a + 2b

e = –2

bonding MO

– C +

+ C – – C + – C + + C –

– C +– C +

+ C + – C –

+ C –

– C ++ C –

+ C –

Fig. 4.17 The p molecular orbitals and p energy levels for a cyclic four-p-orbital system in the simple H€uckel method. The MOs are composed of the basis functions (four p AOs) and the eigenvectors, while the energies of the MOs follow from the eigenvalues (Eq. 4.69). This particular diagram is for the square cyclobutadiene molecule. The paired arrows represent a pair of electrons of opposite spin, in the fully-occupied lowest MO, c1, and the single arrows represents unpaired electrons of the same spin, one in each of the two nonbonding MOs, c2 and c3; the highest p MO, c4, is empty in the neutral molecule

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

4.3.5

133

The Simple H€ uckel Method – Applications

Applications of the SHM are discussed in great detail in several books [21]; here we will deal only with those applications which are needed to appreciate the utility of the method and to smooth the way for the discussion of certain topics (like bond orders and atomic charges) in later chapters. We will discuss: the nodal properties of the MOs; stability as indicated by energy levels and aromaticity (the 4n þ 2 rule); resonance energies; and bond orders and atomic charges. 4.3.5.1

The Nodal Properties of the MOs

A node of an MO is a plane at which, as we proceed along the sequence of basis functions, the sign of the wavefunction changes (Figs. 4.15–4.17). For a given molecule, the number of nodes in the p orbitals increases with the energy. In the two-orbital system (Fig. 4.15), c1 has zero nodes and c2 has one node. In the threeorbital system (Fig. 4.16), c1, c2 and c3 have zero, one and two nodes, respectively. In the cyclic four-orbital system (Fig. 4.17), c1 has zero nodes, c2 and c3, which are degenerate (of the same energy) each have one node (one nodal plane), and c4 has two nodes. In a given molecule, the energy of the MOs increases with the number of nodes. The nodal properties of the SHM p orbitals form the basis of one of the simplest ways of understanding the predictions of the Woodward–Hoffmann orbital symmetry rules [38]. For example, the thermal conrotatary and disrotatary ring closure/opening of polyenes can be rationalized very simply in terms of the symmetry of the highest occupied p MO of the open-chain species. That the highest p MO should dominate the course of this kind of reaction is indicated by more detailed considerations (including extended H€uckel calculations) [38]. Figure 4.18 shows the situation for the ring closure of a 1,3-butadiene to a cyclobutene. The phase (+ or ) of the p HOMO (c2) at the end carbons (the atoms that bond) is opposite on each face, because this orbital has one node in the middle of the C4 chain. You can see this by sketching the MO as the four AOs contributing to it, or even – remembering the node – drawing just the end AOs. For the electrons in c2 to bond, the end groups must rotate in the same sense (conrotation) to bring orbital lobes of the same phase together. Remember that plus and minus phase has nothing to do with electric charge, but is a consequence of the wave nature of electrons (Section 4.2.6): two electron waves can reinforce one another and form a bonding pair if they are “vibrating in phase”; an out-of-phase interaction represents an antibonding situation. Rotation in opposite senses (disrotation) would bring opposite-phase lobes together, an antibonding situation. The mechanism of the reverse reaction is simply the forward mechanism in reverse, so the fact that the thermodynamically favored process is the ring-opening of a cyclobutene simply means that the cyclobutene shown would open to the butadiene shown on heating. Photochemical processes can also be accommodated by the Woodward–Hoffmann orbital symmetry rules if we realize that absorption of a photon creates an electronically excited molecule in which the previous lowest unoccupied MO (LUMO) is

134

4 Introduction to Quantum Mechanics in Computational Chemistry

energy

y4

+ C –

– C +

+ C –

– C +

Draw –

y3

+ C –

– C

+ C –

C +

y2

+

+ –

–

–

3 nodes

+

+

or just

+

+

–

–

2 nodes

y2

+ C

–

–

– C

C

+

C

H

+ +

1 node

H Me

–

y1

+ C

C

–

Me C

C conrotation

no nodes

Me Me +

+ H

H

and

H

– –

H

Me

Me same

Fig. 4.18 The stereochemistry of many reactions is easily predicted from the symmetry of molecular orbitals, usually the highest occupied p MO (p HOMO). In the ring closure of 1,3butadiene to cyclobutene the phase (þ or ) of the HOMO (c2) at the end carbons (the atoms that bond) is such that closure must occur in a conrotatory sense, giving a definite stereochemical outcome. In the example above there is only one product. The reverse process is actually thermodynamically favored, and the cis dimethyl cyclobutene opens to the cis, trans diene. No attempt is made here to show quantitatively the positions of the energy levels or to size the AOs according to their contributions to the MOs

now the HOMO. For more about orbital symmetry and chemical reactions see e.g. the book by Woodward and Hoffmann [38].

4.3.5.2

Stability as Indicated by Energy Levels, and Aromaticity

The MO energy levels obtained from an SHM calculation must be filled with electrons according to the species under consideration. For example, the neutral ethene molecule has two p electrons, so the diagrams of Fig. 4.19a (cf. Fig. 4.15) with one, two and three p electrons, would refer to the cation, the neutral and the anion. We might expect the neutral, with its bonding p orbital c1 full and its antibonding p orbital c2 empty, to be resistant to oxidation (which would require removing electronic charge from the low-energy c1) and to reduction (which would require adding electronic charge to the high-energy c2).

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

a

135

energy antibonding orbitals

nonbonding level

bonding orbitals .+

H2C

CH2

H2C

radical cation

b

CH2

neutral

CH2– radical anion

H2C

energy antibonding orbitals

nonbonding orbitals

.

+ cation

neutral radical

bonding orbitals – anion

c energy antibonding orbitals

nonbonding orbitals

bonding orbitals ++ dication

Fig. 4.19 Filling p MOs with electrons

–– neutral

dianion

136

4 Introduction to Quantum Mechanics in Computational Chemistry

The propenyl (allyl) system has two, three or four p electrons, depending on whether we are considering the cation, radical or anion (Fig. 4.19b; cf. Fig. 4.16). The cation might be expected to be resistant to oxidation, which requires removing an electron from a low-lying p orbital (c1) and to be moderately readily reduced, as this involves adding an electron to the nonbonding p orbital c2, a process that should not be strongly favorable or unfavorable. The radical should be easier to oxidize than the cation, for this requires removing an electron from a nonbonding, rather than a lower-lying bonding, orbital, and the ease of reduction of the radical should be roughly comparable to that of the cation, as both can accommodate an electron in a nonbonding orbital. The anion should be oxidized with an ease comparable to that of the radical (removal of an electron from the nonbonding c2), but be harder to reduce (addition of an electron to the antibonding c3). The cyclobutadiene system (Fig. 4.19c; cf. Fig. 4.17) can be envisaged with, amongst others, two (the dication), four (the neutral molecule) and six p (the dianion) electrons. The predictions one might make for these the behavior of these three species toward redox reactions are comparable to those just outlined for the propenyl cation, radical and anion, respectively (note the analogous occupancy of bonding, nonbonding and antibonding orbitals). The neutral cyclobutadiene molecule is, however, predicted by the SHM to have an unusual electronic arrangement for a diene: in filling the p orbitals, from the lowest-energy one up, one puts electrons of the same spin into the degenerate c2 and c3 in accordance with Hund’s rule of maximum multiplicity. Thus the SHM predicts that cyclobutadiene will be a diradical, with two unpaired electrons of like spin. Actually, more advanced calculations [39] indicate, and experiment confirms, that cyclobutadiene is a singlet molecule with two single and two double C/C bonds. A square cyclobutadiene diradical with four bond order 1.5 C/C bonds would distort to a rectangular, closed-shell (i.e. no unpaired electrons) molecule with two single and two double bonds (Fig. 4.20). This could have been predicted by augmenting the SHM result with a knowledge of the phenomenon known as the Jahn–Teller effect [40]: cyclic systems (and certain others) with an odd number of electrons in degenerate (equal-energy) MOs will distort to remove the degeneracy. What general pattern of molecular orbitals emerges from the SHM? Acyclic p systems (ethene, the propenyl system, 1,3-butadiene, etc.), have MOs distributed singly and evenly on each side of the nonbonding level; the odd-AO systems also have one nonbonding MO (Fig. 4.21). Cyclic p systems (the cyclopropenyl system, cyclobutadiene, the cyclopentadienyl system, benzene, etc.) have a lowest MO and pairs of degenerate MOs, ending with one highest or a pair of highest MOs, depending on whether the number of MOs is even or odd. The total number of MOs is always equal to the number of basis functions, which in the SHM is, for organic polyenes, the number of p orbitals (Fig. 4.21). The pattern for cyclic systems can be predicted qualitatively simply by sketching the polygon, with one vertex down, inside a circle (Fig. 4.22). If the circle is of radius 2|b| the energies can even be calculated by trigonometry [41]. It follows from this pattern that cyclic species (not necessarily neutral) with 2, 6, 10, ... p electrons have filled p MOs and might be expected to show particular stability, analogously to the filled AOs of the

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

137

bond order 1.5 bond order 1 i.e. square

Jahn-Tellertype distotion

bond order 2

(pseudo-Jahn-Teller distortion) rectangular

energy

nonbonding level

Fig. 4.20 Cyclic systems with degenerate energy levels tend to undergo a geometric distortion to remove the degeneracy, a consequence of the Jahn–Teller theorem

unreactive noble gases (Fig. 4.23). The archetype of such molecules is, of course, benzene, and the stability is associated with the general collection of properties called aromaticity [17]. These results, which were first perceived by H€uckel [19] (1931–1937), are summarized in a rule called the 4n + 2 rule or H€uckel’s rule, although the 4n + 2 formulation was evidently actually due to Doering and Knox (1954) [42]. This says that cyclic arrays of sp2-hybridized atoms with 4n + 2 p electrons are characteristic of aromatic molecules; the canonical aromatic molecule benzene with six p electrons corresponds to n ¼ 1. For neutral molecules with formally fully conjugated perimeters this amounts to saying that those with an odd number of C/C double bonds are aromatic and those with an even number are antiaromatic (see Section 4.3.5.3). H€ uckel’s rule has been abundantly verified [17] notwithstanding the fact that the SHM, when applied without regard to considerations like the Jahn–Teller effect (see above) incorrectly predicts 4n species like cyclobutadiene to be triplet diradicals. The H€ uckel rule also applies to ions; for example, the cyclopropenyl system two p electrons, the cyclopropenyl cation, corresponds to n ¼ 0, and is strongly aromatic. Other aromatic species are the cyclopentadienyl anion (six p electrons, n ¼ 1; H€ uckel predicted the enhanced acidity of cyclopentadiene) and the cycloheptatrienyl cation. Only reasonably planar species can be expected to provide the AO overlap need for cyclic electron delocalization and aromaticity, and care is needed in applying the rule. Electron delocalization and aromaticity within the SHM have recently been revisited [43].

138

4 Introduction to Quantum Mechanics in Computational Chemistry

a- 2b

a- b

0

nonbonding level

a+ b

a+ 2b

C2

C3

C4

C5

C6

acyclic species

a- 2b

a- b

0

nonbonding level

a+ b

a+ 2b

C3

C4

C5

C6

C7

cyclic species

Fig. 4.21 The MO pattern for acyclic and cyclic p systems, as predicted by the simple H€ uckel method

4.3.5.3

Resonance Energies

The SHM permits the calculation of a kind of stabilizing energy, or, more accurately, an energy that reflects the stability of molecules. This energy is calculated by comparing the total electronic energy of the molecule in question with that of a reference compound, as shown below for the propenyl systems, cyclobutadiene, and the cyclobutadiene dication.

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel a-2b

a – 2b

a – 1.618b

a-b

a-b .

.

a

a – 1.618 b

a - b

.

a

a+2b

a-b nonbonding level, a

.

a + 0.618 b

a + 0.618 b a+2b

139

a+b

a + b a+2b

a+2b

Fig. 4.22 A useful mnemonic for getting the simple H€ uckel method pattern for cyclic p systems. Setting the radius of the circle at 2|b|, the energy separations from the nonbonding level can even be calculated by trigonometry a–2b

a–2b a–b

a – 1.618 b

a–b a

a – 1.618 b

a–b

a–b

a

a a + 0.618 b

a+2b

a+2b

+

a + 0.618 b

a+b

a+b

a+2b

a+2b

+

–

Fig. 4.23 H€uckel’s rule says that cyclic p systems with 4n þ 2 p electrons (n ¼ 0, 1, 2, . . .; 4n þ 2 ¼ 2, 6, 10, . . .) should be especially stable, since they have all bonding levels full and all antibonding levels empty. The special stability is usually equated with aromaticity. Shown here are the cyclopropenyl cation, the cyclobutadiene dication, the cyclopentadienyl anion, and benzene; formal structures are given for these species – the actual molecules do not have single and double bonds, but rather electron delocalization makes all C/C bonds the same

The propenyl cation, Fig. 4.19b; cf. Fig. 4.16. If we take the total p electronic energy of a molecule to be simply the number of electrons in a p MO times the energy level of the orbital, summed over occupied orbitals (a gross approximation, as it ignores interelectronic repulsion), then for the propenyl cation Ep ðprop: cationÞ ¼ 2ða þ 1:414bÞ ¼ 2a þ 2:828b We want to compare this energy with that of two electrons in a normal molecule with no special features (the propenyl cation has the special feature of a formally empty p orbital adjacent to the formal C/C double bond), and we choose neutral ethene for our reference energy (Fig. 4.15) Ep ðreferenceÞ ¼ 2ða þ bÞ ¼ 2a þ 2b The stabilization energy is then E(stab, cation) ¼ Ep ðprop: cationÞ  Ep ðreferenceÞ ¼ ð2a þ 2:828 bÞ  ð2a þ 2bÞ ¼ 0:828b

140

4 Introduction to Quantum Mechanics in Computational Chemistry

Since b is negative, the p-electronic energy of the propenyl cation is calculated to be below that of ethene: providing an extra, formally empty p orbital for the electron pair causes the energy to drop. Actually, resonance energy is usually presented as a positive quantity, e.g. “100 kJ mol1”. We can interpret this as 100 kJ mol1 below a reference system. To avoid a negative quantity in SHM calculations like these, we can use |b| instead of b. The propenyl radical, Fig. 4.16. The total p electronic energy by the SHM is Ep ðprop: radicalÞ ¼ 2ða þ 1:414bÞ þ a ¼ 3a þ 2:828b For the reference energy we use one ethene molecule and one nonbonding p electron (like the electron in a methyl radical): Ep ðreferenceÞ ¼ ð2a þ 2bÞ þ a ¼ 3a þ 2b The stabilization energy is then E(stab, radical) ¼ Ep ðprop: radicalÞ  Ep ðreferenceÞ ¼ ð3a þ 2:828bÞ  ð3a þ 2bÞ ¼ 0:828b The propenyl anion. An analogous calculation (cf. Fig. 4.16, with four electrons for the anion) gives E(stab anion) ¼ Ep ðprop: anionÞ  Ep ðreferenceÞ ¼ ð4a þ 2:828bÞ  ð4a þ 2bÞ ¼ 0:828b Thus the SHM predicts that all three propenyl species will be lower in energy than if the p electrons were localized in the formal double bond and (for the radical and anion) in one p orbital. Because this lower energy is associated with the ability of the electrons to spread or be delocalized over the whole p system, what we have called E(stab) is often denoted as the delocalization energy, and designated ED. Note that ER (resonance energy, or ED, delocalization energy) is always some multiple of b (or is zero). Since electron delocalization can be indicated by the familiar resonance symbolism the H€ uckel delocalization energy is often equated with resonance energy, and designated ER. The accord between calculated delocalization and the ability to draw resonance structures is not perfect, as indicated by the next example. Cyclobutadiene (Fig. 4.17). The total p electronic energy is Ep ðcyclobutadieneÞ ¼ 2ða þ 2b) + 2 a ¼ 4a þ 4b Using two ethene molecules as our reference system: Ep ðreferenceÞ ¼ 2a þ 2b

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

141

and so for E(stab) (¼ ED or ER) we get E(stap, cyclobutadiene) ¼ Ep ðcyclobutadieneÞ  Ep ðreferenceÞ ¼ ð4a þ 4bÞ  ð4a þ 4bÞ ¼ 0 Cyclobutadiene is predicted by this calculation to have no resonance energy, although we can readily draw two “resonance structures” exactly analogous to the Kekule´ structures of benzene. The SHM predicts a resonance energy of 2b for benzene. Equating 2|b| with the commonly-quoted resonance energy of 150 kJ mol1 (36 kcal mol1) for benzene gives a value of 75 kJ mol1 for |b|, but this should be taken with more than a grain of salt, for outside a closely related series of molecules, b has little or no quantitative meaning [44]. However, in contrast to the failure of simple resonance theory in predicting aromatic stabilization (and other chemical phenomena) [45], the SHM is quite successful. The cyclobutadiene dication (cf. Fig. 4.17). The total p electronic energy is Ep ðdicationÞ ¼ 2ða þ 2bÞ ¼ 2a þ 4b Using one ethene molecule as the reference: Ep ðreferenceÞ ¼ 2a þ 2b and so E(stab, dication) ¼ Ep ðdicationÞ  Ep ðreferenceÞ ¼ ð2a þ 4bÞ  ð2a þ 2bÞ ¼ 2b Thus the stabilization energy calculation agrees with the deduction from the disposition of filled MOs (i.e. with the 4n þ 2 rule) that the cyclobutadiene dication should be stabilized by electron delocalization, which is in some agreement with experiment [46]. More sophisticated calculations indicate that cyclic 4n systems like cyclobutadiene (where planar; cyclooctatetraene, for example, is buckled by steric factors and is simply an ordinary polyene) are actually destabilized by p electronic effects: their resonance energy is not just zero, as predicted by the SHM, but less than zero. Such systems are antiaromatic [17, 46]. 4.3.5.4

Bond Orders

The meaning of this term is easy to grasp in a qualitative, intuitive way: an ideal single bond has a bond order of one, and ideal double and triple bonds have bond orders of two and three, respectively. Invoking Lewis electron-dot structures, one might say that the order of a bond is the number of electron pairs being shared between the two bonded atoms. Calculated quantum mechanical bond orders should

142

4 Introduction to Quantum Mechanics in Computational Chemistry

be more widely applicable than those from the Lewis picture, because electron pairs are not localized between atoms in a clean pairwise manner; thus a weak bond, like a hydrogen bond or a long single bond, might be expected to have a bond order of less than one. However, there is no unique definition of bond order in computational chemistry, because there seems to be no single, correct method to assign electrons to particular atoms or pairs of atoms [47]. Various quantum mechanical definitions of bond order can be devised [48], based on basis-set coefficients. Intuitively, these coefficients for a pair of atoms should be relevant to calculating a bond order, since the bigger the contribution two atoms make to the wavefunction (whose square is a measure of the electron density; Section 4.2.6), the bigger should be the electron density between them. In the SHM the order of a bond between two atoms Ai and Bj is defined as Bi;j ¼ 1 þ

X

nci cj

(4.70)

all occ

Here the 1 denotes the single bond of the ubiquitous spectator s bond framework, which is taken as always contributing a s bond order of unity. The other term is the p bond order; its value is obtained by summing over all the occupied MOs the number of electrons n in each of these MOs times the product of the c’s of the two atoms for each MO. This is illustrated in these examples: Ethene. The occupied orbital is c1, which has 2 electrons), and the coefficients of c1 and c2 for this orbital are 0.707, 0.707 (Eq. 4.66). Thus Bi;j ¼ 1 þ

X

nci cj ¼ 1 þ 2ð0:707Þ0:707 ¼ 1 þ 1:000 ¼ 2:000

all occ

which is reasonable for a double bond. The order of the s bond is 1 and that of the p bond is 1. The ethene radical anion. The occupied orbitals are c1, which has 2 electrons, and c2, which has 1 electron; the coefficients of c1 and c2 for c1 are 0.707, 0.707 and for c2, 0.707, 0.707 (Eq. 4.66). Thus Bi;j ¼ 1 þ

X

nci cj ¼ 1 þ 2ð0:707Þ0:707 þ 1ð0:707Þð0:707Þ

all occ

¼ 1 þ 1  0:500 ¼ 1:500 The p bond order of 0.500 (1.500  s bond order) accords with two electrons in the bonding MO and one electron in the antibonding orbital.

4.3.5.5

Atomic Charges

In an intuitive way, the charge on an atom might be thought to be a measure of the extent to which the atom repels or attracts a charged probe near it, and to be

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

143

measurable from the energy it takes to bring a probe charge from infinity up to near the atom. However, this would tell us the charge at a point outside the atom, for example a point on the van der Waals surface of the molecule, and the repulsive or attractive forces on the probe charge would be due to the molecule as a whole. Although atomic charges in molecules are generally considered to be experimentally unmeasurable (but see Chapter 5, section 5.5.4, Charges and bond orders), chemists find the concept very useful (thus calculated charges are used to parameterize molecular mechanics force fields – Chapter 3), and much effort has gone into designing various definitions of atomic charge [48, 49]. Intuitively, the charge on an atom should be related to the basis set coefficients of the atom, since the more the atom contributes to a multicenter wavefunction (one with contributions from basis functions on several atoms), the more it might be expected to lose electronic charge by delocalization into the rest of the molecule (cf. the discussion of bond order above). In the SHM the charge on an atom Ai is defined as (cf. Eq. 4.70) qi ¼ 1 

X

nc2i

(4.71)

all occ

The summation term is the charge density, and is a measure of the electronic charge on the molecule due to the p electrons. For example, having no p electrons (an empty p orbital, formally a cationic carbon) would mean a p electron charge density of zero; subtracting this from unity gives a charge on the atom of þ1. Again, having two p electrons in a p orbital would mean a p electron charge density of 2 on the atom; subtracting this from unity gives a charge on the atom of 1 (a filled p orbital, formally an anionic carbon). The application of Eq. 4.71 will be illustrated using methylenecyclopropene (Fig. 4.24).

4.3.5.6

Methylenecyclopropene X

q1 ¼ 1 

h i nc21 ¼ 1  2ð0:282Þ2 þ 2ð0:815Þ2 ¼ 1  1:487 ¼ 0:487

all occ

q2 ¼ 1 

X

h i nc22 ¼ 1  2ð0:612Þ2 þ 2ð0:254Þ2 ¼ 1  0:878 ¼ 0:122

all occ

q3 ¼ q4 ¼ 1 

X

h i nc23 ¼ 1  2ð0:523Þ2 þ 2ð0:368Þ2 ¼ 1  0:817 ¼ 0:182

all occ

The results of this charge calculation are summarized in Fig. 4.24; the negative charge on the exocyclic carbon and the positive charges on the ring carbons are in accord with the resonance picture (Fig. 4.24), which invokes a contribution from the aromatic cyclopropenyl cation [50]. Note that the charges sum to (essentially) zero, as they must for a neutral molecule (the hydrogens, which actually also carry

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4 Introduction to Quantum Mechanics in Computational Chemistry

a - 2b

c = 0.815 in y 2 0.282 in y 1

a-b

c = 0.254 in y 2

q 1 = – 0.487 – q 2 = 0.122

0.612 in y 1 +

y2 a+b

a + 2b

a + 0.311 b

c = – 0.368 in y 2

q 3 = 0.182

0.523 in y 1

y1 a + 2.1700

Fig. 4.24 The SHM charges on the atoms of a molecule can be calculated from the number of electrons in each occupied MO and the coefficients of these MOs. The predicted dipolar nature of methylenecyclopropene has been ascribed to a cyclopropenyl-cation-like resonance contributor

charges, have been excluded from consideration here). A high-level calculation places a total charge (carbon plus hydrogen) – albeit defined in a different way – of 0.37 on the CH2 group and þ0.37 on the ring (cf. 0.487 and +0.487 for the exocyclic carbon and the ring carbons in the SHM calculation). There are many other applications of the SHM [21c, e] including fairly recent and perhaps unexpected ones such as correlation with UV solvent shifts [51] and even physicochemical properties [52].

4.3.6

Strengths and Weaknesses of the Simple H€ uckel Method

4.3.6.1

Strengths

The SHM has been extensively used to correlate, rationalize, and predict many chemical phenomena, having been applied with surprising success to dipole moments, ESR spectra, bond lengths, redox potentials, ionization energies, UV and IR spectra, aromaticity, acidity/basicity, and reactivity, and specialized books on the SHM should be consulted for details [21, 22]. The method will give probably give some insight into any phenomenon that involves predominantly the p electron systems of conjugated molecules. The SHM may have been underrated [53] and reports of its death are probably exaggerated. However, the SHM is not used very much in research nowadays, partly because more sophisticated p electron

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

145

approaches like the PPP method (Section 6.2.2) are available, but mainly because of the phenomenal success of all-valence-electron semiempirical methods (Chapter 6), which are applicable to quite large molecules, and of the increasing power of all-electron ab initio (Chapter 5) and DFT (Chapter 7) methods.

4.3.6.2

Weaknesses

The defects of the SHM arise from the fact that it treats only p electrons, and these only very approximately. The basic H€ uckel method described here has been augmented in an attempt to handle non-p substituents, e.g. alkyl groups, halogen groups, etc., and heteroatoms instead of carbon. This has been done by treating the substituents as p centers and embodying empirically altered values of a and b, so that in the Fock matrix values other than 1 and 0 appear. However, the values of these modified parameters that have been employed vary considerably [54], which tends to diminish one’s confidence in their reliability. The approximations in the SHM are its peremptory treatment of the overlap integrals S (Section 4.3.4, discussion in connection with Eqs. 4.55), its drastic truncation of the possible values of the Fock matrix elements into just a, b and 0 (Section 4.3.4, discussion in connection with Eqs. 4.61), its complete neglect of electron spin, and its glossing over (although not exactly ignoring) interelectronic repulsion by incorporating this into the a and b parameters. The overlap integrals S are divided into just two classes: Z fi fj dv ¼ Sij ¼ 1 or 0 depending on whether the orbitals on the atoms i and j are on the same or different atoms. This approximation, as explained earlier, reduces the matrix form of the secular equations to standard eigenvalue form HC ¼ C« (Eq. 4.59), so that the Fock matrix can (after giving its elements numerical values) be diagonalized without further ado (the ado is explained in Section 4.4.1, in connection with the extended H€ uckel method). In the older determinant, as opposed to matrix, treatment (Section 4.3.7), the approximation greatly simplifies the determinants. In fact, however, the overlap integral between adjacent carbon p orbitals is ca. 0.24 [55]. Setting the Fock matrix elements equal to just a, b and 0: Setting Z

^ j dv ¼ Hij ¼ a; b or 0 fi Hf

depending on whether the orbitals on the atoms i and j are on the same, adjacent or further-removed atoms is an approximation, because all the Hii terms are not the same, and all the adjacent-atom Hij terms are not the same either; these energies depend on the environment of the atom in the molecule; for example, atoms in the

146

4 Introduction to Quantum Mechanics in Computational Chemistry

middle of a conjugated chain should have different Hii and Hij parameters than ones at the end of the chain. Of course, this approximation simplifies the Fock matrix (or the determinant in the old determinant method, Section 4.3.7). The neglect of electron spin and the deficient treatment of interelectronic repulsion is obvious. In the usual derivation (Section 4.3.4): in Eq. 4.40 the integration is carried out with respect to only spatial coordinates (ignoring spin coordinates; contrast ab initio theory, Section 5.2), and in calculating p energies (Section 4.3.5.3) we simply took the sum of the number of electrons in each occupied MO times the energy level of the MO. However, the energy of an MO is the energy of an electron in the MO moving in the force field of the nuclei and all the other electrons (as pointed out in Section 4.3.4, in explaining the matrices of Eqs. 4.55). If we calculate the total electronic energy by simply summing MO energies times occupancy numbers, we are assuming, wrongly, that the electron energies are independent of one another, i.e. that the electrons do not interact. An energy calculated in this way is said to be a sum of one-electron energies. The resonance energies calculated by the SHM can thus be only very rough, unless the errors tend to cancel in the subtraction step, which in fact probably occurs to some extent (this is presumably why the method of Hess and Schaad for calculating resonance energies works so well [53]). The neglect of electron repulsion and spin in the usual derivation of the SHM is discussed in reference [30a].

4.3.7

The Determinant Method of Calculating the H€ uckel c’s and Energy Levels

An older method of obtaining the coefficients and energy levels from the secular equations (Eqs. 4.49 for a two-basis-function system) utilizes determinants rather than matrices. The method is much more cumbersome than the matrix diagonalization approach of Section 4.3.4, but in the absence of cheap, readily-available computers (matrix diagonalization is easily handled by a personal computer) its erstwhile employment may be forgiven. It is outlined here because traditional presentations of the SHM [21] use it. Consider again the secular equations 4.49: ðH11  HS11 Þc1 þ ðH12  ES12 Þc2 ¼ 0 ðH21  HS21 Þc1 þ ðH22  ES22 Þc2 ¼ 0 By considering the requirements for nonzero values of c1 and c2 we can find how to calculate the c’s and the molecular orbital energies (since the coefficients are weighting factors that determine how much each basis function contributes to the MO, zero c’s would mean no contributions from the basis functions and hence no MOs; that would not be much of a molecule). Consider the system of linear equations

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147

A11 x1 þ A12 x2 ¼ b1 A21 x1 þ A22 x2 ¼ b2 Using determinants:  A12   A22  x1 ¼ D    A11 b1       A21 b2  x2 ¼ D    A11 A12    D¼   A21 A22    b1    b2

where D is the determinant of the system. If b1 ¼ b2 ¼ 0 (the situation in the secular equations), then in the equations for x1 and x2 the numerator is zero, and so x1 ¼ 0/D and x2 ¼ 0/D. The only way that x1 and x2 can be nonzero in this case is that the determinant of the system be zero, i.e. D¼0 for then x1 ¼ 0/0 and x2 ¼ 0/0, and 0/0 can have any finite value; mathematicians call it indeterminate. This is easy to see: Let 0 ¼a 0 then a0¼0 which is true for any finite value of a. So for the secular equations the requirement that the c’s be nonzero is that the determinant of the system be zero:   H  ES 11  11 D¼  H21  ES21

 H12  ES12  ¼0 H22  ES22 

(4.72)

148

4 Introduction to Quantum Mechanics in Computational Chemistry

Equation 4.74 can be generalized to n basis functions (cf. the matrix of Eq. 4.62):   H11  ES11 H12  ES12    H1 n  ES1n    H21  ES21 H22  ES22    H2 n  ES2 n   .. .. ..  .  . .   Hn1  ESn1 Hn2  ESn2    Hnn  ESnn

     ¼0    

(4.73)

If we invoke the SHM simplification of orthogonality of the S integrals (pp. 37–39), then Sii ¼ 1 and Sij ¼ 0 and Eq. 4.73 becomes          

H11  E

H12



H22  E    .. .  Hn2 

H21 .. . Hn1

    H2n  ¼0  ..  .  Hnn  E  H1n

(4.74)

Substituting a, b and 0 for the appropriate H’s (Eqs. 4.61a, b, c) we get     aE b ... 0     a  E ... 0  b ¼0  (4.75)  . . . .. .   .. . . . .   0 0 ... a  E  The diagonal terms will always be a  E, but the placement of b and 0 will depend on which i, j terms are adjacent and which are further-removed, which depends on the numbering system chosen (see below). Since multiplying or dividing a determinant by a number is equivalent to multiplying or dividing the elements of one row or column by that number (Section 4.3.3), multiplying both sides of Eq. 4.75 by 1/b n times, i.e. by (1/b)n gives     ð a  EÞ=b 1 ... 0     ða  EÞ=b . . . 0  1 ¼0  (4.76)  . . . .. .   .. . . . .   0 0 . . . ða  E Þ=b  Finally, if we define (aE)/b ¼ x, we get          

 x 1 . . . 0   1 x ... 0  ¼0 ..  .. .. . . ... .  0 0 ... x 

(4.77)

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel

149

The diagonal terms are always x but the off-diagonal terms, 1 for adjacent and 0 for nonadjacent orbital pairs, depend on the numbering (which does not affect the results: Fig. 4.25). Any specific determinant of the type in Eq. 4.77 can be expanded into a polynomial of order n (where the determinant is of order n  n), making Eq. 4.77 yield polynomial equation: xn þ a1 xn1 þ a2 xn2 þ    an ¼ 0

(4.78)

The polynomial can be solved for x and then the energy levels can be found from (a  E)/b ¼ x, i.e. from E ¼ a  bx

(4.79)

The coefficients can then be calculated from the energy levels by substituting the E’s into one of the secular equations, finding the ratio of the c’s, and normalizing to get the actual c’s. An example will indicate how the determinant method can be implemented. Consider the propenyl system. In the secular determinant the i,i-type interactions will be represented by x, adjacent i, j-type interactions by 1 and nonadjacent i, j-type interactions by 0. For the determinantal equation we can write (Fig. 4.25)

1

3

2

2

1

1

1

1

x

1

1

1

x

x

1

0

1

x

1

0

1

x

x

1

1

1

x

0

1

0

x

= x 3 – 2x

3

1

2

x

3

= x 3 – 2x

Fig. 4.25 The determinants corresponding to different numbering patterns can seem to differ, but on expansion they give the same polynomial

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4 Introduction to Quantum Mechanics in Computational Chemistry

  x 1 0     1 x 1¼0   0 1 x

(4.80)

(compare this with the Fock matrix for the propenyl system). Solving this equation (see Section 4.3.3):        x 1 0 1 x 1 1 x 1           þ 1   1  1 x 1  ¼ x 0 1 0 x 1 x   0 1 x

    

¼ xðx2  1Þ  ðx  0Þ þ 0 ¼ x3  x  x ¼ x3  2x ¼ 0

(4.81)

This cubic can be factored (but in general polynomial equations require numerical approximation methods): xðx2  2Þ ¼ 0

so

x¼0

and x2  2 ¼ 0

or x ¼ 

p

2

From (a  E)/b ¼ x, E ¼ a  xb and x ¼ 0 leads to E ¼ a p p x ¼ þ 2 leads to E ¼ a  2b p p x ¼  2 leads to E ¼ a þ 2b pffi So we get the same energy levels as from matrix diagonalization ( 2 ¼ 1.414). To find the coefficients we substitute the energy levels into the secular equations; for the propenyl system these are, projecting from the secular equations for a two-orbital system, Eqs. 4.49: ðH11  ES11 Þc1 þ ðH12  ES12 Þc2 þ ðH13  ES13 Þc3 ¼ 0 ðH21  ES21 Þc1 þ ðH22  ES22 Þc2 þ ðH23  ES23 Þc3 ¼ 0

(4.82)

ðH31  ES31 Þc1 þ ðHS32  ES32 Þc2 þ ðH33  ES33 Þc3 ¼ 0 These can be simplified (Eqs. 4.57, 4.61) to ða  EÞc1 þ bc2 þ 0c3 ¼ 0 bc1 þ ða  EÞc2 þ bc3 ¼ 0

(4.83)

0c1 þ bc2 þ ða  EÞc3 ¼ 0 For the energy level E ¼ a þ secular equation we get

p

2b (MO level 1, c1), substituting into the first

4.3 The Application of the Schr€ odinger Equation to Chemistry by H€ uckel



p

2bc11 þ bc21 ¼ 0;

so c21 =c11 ¼

p

151

2

(Recall the cij notation; c11 is the coefficient for atomp1ffi in c1, c21 is the coefficient for atom 2 in c1, etc.). Substituting E ¼ a + 2b into the second secular equation we get  bc11 þ bc31 ¼ 0;

so c11 ¼ c31

We now have the relative values of the c’s: p c11 =c11 ¼ 1; c21 =c11 ¼ 2; c31 =c11 ¼ 1

(4.84)

To find the actual values of the c’s, we utilize the fact that the MO (we are talking about MO level 1, c1) must be normalized: Z c21 dv ¼ 1 (4.85) Now, from the LCAO method c1 ¼ c11 f1 þ c21 f2 þ c31 f3

(4.86)

Therefore c21 ¼ c211 f21 þ c221 f22 þ c231 f23 þ 2c11 c21 f1 f2 þ 2c11 c31 f1 f3 þ 2c21 c31 f2 f3

(4.87)

So from Eq. 4.87, and recalling that in the SHM we pretend that the basis functions f are orthonormal, i.e. that Sij ¼ dij, we get Z c21 dv ¼ c211 þ c221 þ c231 ¼ 1 (4.88) Using the ratios of the c’s from Eq. 4.84: c211 c221 c231 1 þ þ ¼ c211 c211 c211 c211 i.e. 12 þ

pffiffiffi2 1 2 þ 12 ¼ 2 c11

and so c11 ¼

1 2

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4 Introduction to Quantum Mechanics in Computational Chemistry

and c21 ¼

pffiffiffi 1 2 c11 ¼ pffiffiffi 2

and c31 ¼ c11 ¼ 1 By substituting into the secular equations 4.83 the E values for c2 and c3 we could find the ratios of the c’s for c2 and c3 and with the aid of the orthonormalization equation analogous to Eq. 4.88 we could get the actual values of c12, c22, c32 and c13, c23, and c33. Although this somewhat clumsy way of finding the c’s from the energy levels was streamlined (see e.g. [21d]), the determinant method has been replaced by matrix diagonalization implemented in a computer program.

4.4

The Extended H€ uckel Method

4.4.1

Theory

In the simple H€ uckel method, as in all modern molecular orbital methods, a Fock matrix is diagonalized to give coefficients (which with the basis set give the wavefunctions of the molecular orbitals) and energy levels (i.e. molecular orbital energies). The SHM and the extended H€ uckel method (EHM, extended H€uckel theory, EHT) differ in how the elements of the Fock matrix are obtained and how the overlap matrix is treated. The EHM was popularized and widely applied by Hoffmann28 [56], although earlier work using the approach had been done by Wolfsberg and Helmholz [57]. We now compare point by point the SHM and the EHM. 4.4.1.1

Simple H€ uckel Method

1. Basis set is limited to p orbitals. Each element of the Fock matrix H is an integral that represents an interaction between two orbitals. The orbitals are in almost all cases a set of p orbitals (usually carbon 2p) supplied by an sp2 framework, with the p orbital axes parallel to one another and perpendicular to the plane of the framework. In other words, the set of basis orbitals – the basis set – is limited (in 28

Roald Hoffmann, born Zloczow, Poland, 1937. Ph.D. Harvard, 1962, Professor, Cornell. Nobel Prize 1981(shared with Kenichi Fukui; Section 7.3.5) for work with organic chemist Robert B. Woodward, showing how the symmetry of molecular orbitals influences the course of chemical reactions (the Woodward–Hoffmann rules or the conservation of orbital symmetry). Main exponent of the extended H€ uckel method. He has written poetry, and several popular books on chemistry.

4.4 The Extended H€uckel Method

153

the great majority of cases) to pz orbitals (taking the framework plane, i.e. the molecular plane, to be the xy plane). 2. Orbital interaction energies are limited to a, b and 0. The Fock matrix orbital interactions are limited to a, b and 0, depending on whether the Hij interaction is, respectively i,i, adjacent, or further-removed. The value of b does not vary smoothly with the separation of the orbitals, although logically it should decrease continuously to zero as the separation increases. 3. Fock matrix elements are not actually calculated. The Fock matrix elements are not any definite physical quantities, but rather energy levels relative to a in units of |b|, making them 0 or 1. One can try to estimate a and b, but the SHM does not define them quantitatively. 4. Overlap integrals are limited to 1 or 0. We pretend that the overlap matrix S is a unit matrix, by setting Sij ¼ dij. This enables us to simplify HC ¼ SC« (Eq. 4.54) to the standard eigenvalue form HC ¼ C« (Eq. 4.59) and so H ¼ C«C1, which is the same as saying that the SHM Fock matrix is directly diagonalized to give the c’s and e’s. Now compare these four points with the corresponding features of the EHM:

4.4.1.2

Extended H€ uckel Method

1. All valence s and p orbitals are used in the basis set. As in the SHM each element of the Fock matrix is an integral representing an interaction between two orbitals; however, in the EHM the basis set is not just a set of 2pz orbitals but rather the set of valence-shell orbitals of each atom in the molecule (the derivation of the secular equations says nothing about what kinds of orbitals we are considering). Thus each hydrogen atom contributes a 1s orbital to the basis set and each carbon atom a 2s and three 2p orbitals. Lithium and beryllium, although they have no 2p electrons, are assigned a 2s and three 2p orbitals (experience shows that this works better than omitting these basis functions) so the atoms from lithium to fluorine each contribute a 2s and three 2p orbitals. A basis set like this, which uses the normal valence orbitals of atoms, is called a minimal valence basis set. 2. Orbital interaction energies are calculated and vary smoothly with geometry. The EHM Fock matrix orbital interactions Hij are calculated in a way that depends on the distance apart of the orbitals, so their values vary smoothly with orbital separation. 3. Fock matrix elements are actually calculated. The EHM Fock matrix elements are calculated from well-defined physical quantities (ionization energies) with the aid of well-defined mathematical functions (overlap integrals), and so are closely related to ionization energies and have definite quantitative values. 4. Overlap integrals are actually calculated. We do not in effect ignore the overlap matrix, i.e. we do not set it equal to a unit matrix. Instead, the elements of the overlap matrix are calculated, each Sij depending on the distance apart of the

154

4 Introduction to Quantum Mechanics in Computational Chemistry

atoms i and j, which has the important consequence that the S values depend on the geometry of the molecule. Since S is not taken as a unit matrix, we cannot go directly from HC ¼ SC« to HC ¼ C« and thus we cannot simply diagonalize the EHM Fock to get the c’s and e’s. These four points are elaborated on below. 1. Use of a minimal valence basis set in the EHM is more realistic than treating just the 2pz orbitals, since all the valence electrons in a molecule are likely to be involved in determining its properties. Further, the SHM is largely limited to p systems, i.e. to alkenes and aromatics and derivatives of these with attached p electron groups, but the EHM, in contrast, can in principle be applied to any molecule. The use of a minimal valence basis set makes the Fock matrix much larger than in the “corresponding” SHM calculation. For example in an SHM calculation on ethene, only two orbitals are used, the 2pz on C1 and the 2pz on C2E, and the SHM Fock matrix is (using the compact Dirac notation D   R ^ j dv fi H^fj ¼ fi Hf



! ^ 1 ð2pz Þ ^ 2 ð2pz Þ C1 ð2pz ÞjHjC C1 ð2pz ÞjHjC HðSHMÞ ¼



^ 1 ð2pz Þ ^ 2 ð2pz Þ C2 ð2pz ÞjHjC C2 ð2pz ÞjHjC
 0 1 ¼ 2  2 matrix 1 0

(4.89)

To write down the EHM Fock matrix, let us label the valence orbitals like this: H1 ð1sÞ f1

C1 ð2sÞ f5

C1 ð2px Þ f7

C1 ð2py Þ f9

H2 ð1sÞ f2

C2 ð2sÞ f6

C2 ð2px Þ f8 H3 ð1sÞ f3

C2 ð2py Þ f10

C1 ð2px Þ f11 C2 ð2pz Þ f12

H4 ð1sÞ f4 Then 0

^ 1 f1 jHjf B

B ^ 1 B f2 jHjf B HðEHMÞ ¼ B .. B B . @

^ 1 f12 jHjf





^ 2 f1 jHjf ^ 2 f2 jHjf



.. . ^ 2 f12 jHjf

...



^ 12 f1 jHjf

1

C ^ 12 C f2 jHjf C C C .. C C ... . A

^ 12 . . . f12 jHjf ...

12  12 matrix



(4.90)

4.4 The Extended H€uckel Method

H

155

H

H

C

H

C

C

H

C

H

H

H

The simple Hückel method basis set for ethene. Each carbon has one 2p basis function. C2H4 has two basis functions

The extended Hückel method basis set for ethene. Each carbon has one 2s and three 2p basis functions. Each H has one 1s basis function. C2H4 has 12 basis functions.

Fig. 4.26 The simple H€ uckel method normally uses only one basis function per “heavy atom”: only one 2p orbital on each carbon, oxygen, nitrogen, etc., ignoring the hydrogens. The extended H€uckel method uses for each carbon, oxygen, nitrogen, etc., a 2s and three 2p orbitals, and for each hydrogen a 1s orbital. This is called a minimal valence basis set

The SHM and EHM basis sets are shown in Fig. 4.26. 2. The EHM Fock matrix interactions i,j do not have just two values (a or b) as in the SHM, but are functions of the orbitals (the basis functions) fi and fj and of the separation of these orbitals, in D E (3) below.

as explained ^ i and fi Hf ^ j are calculated (rather than set 3. The EHM matrix elements fi Hf equal to 0 or 1), although the calculation is a simple one using overlap integrals and experimental ionization energies; in ab initio calculations (Chapter 5) and more advanced semiempirical calculations (Chapter 6), the mathematical form of Hˆ taken into account. The i,i-type interactions are taken as being proportional to the negative of the ionization energy [58] of the orbital fi and the i,j-type interactions as being proportional to the overlap integral between fi and fj and the negative of the average of the ionization energies Ii and Ij of fi and fj (the negative of the orbital ionization energy is the energy of an electron in the orbital, compared to the zero of energy of the electron and the ionized species infinitely separated and at rest): D

^ i ¼ Ii fi Hf

(4.91)

E ^ j ¼  1 KSij ðIi þ Ij Þ fi Hf 2

(4.92)

A proportionality constant K of about 2 is commonly used. For H(1s), C(2s) and C(2p), experiment shows I ðH(1s)Þ ¼ 13:6 eV,

I ðC(2s)Þ ¼ 20:8 eV,

I ðC(2p)Þ ¼ 11:3 eV

(4.93)

The overlap integrals are calculated using Slater-type (Section 5.3.2) functions for the basis functions, e.g.

156

4 Introduction to Quantum Mechanics in Computational Chemistry


3 12 z fð1sÞ ¼ 1 expðz1 jr  R1s jÞ p
fð2sÞ ¼

12

z52 96p


 z2 jr  R2s1 j jr  R2s j exp 2

(4.94)

(4.95)

where the parameters z depend on the particular atom (H, C, etc.) and orbital (1s, 2s, etc.). The variable r  R is the distance of the electron from the atomic nucleus on which the function is centered; r is the vector from the origin of the Cartesian coordinate system to the electron, and R is the vector from the origin to the nucleus on which the basis function is centered: h i12 jr  RA j ¼ ðx  xA Þ2 þ ðy  yA Þ2 þ ðz  zA Þ2

(4.96)

where (xA, yA, zA) are the coordinates of the nucleus bearing the Slater function. The Slater function is thus a function of three variables x,y,z and depends parametrically on the location (xA, yA, zA) of the nucleus A on which it is centered. The Fock matrix elements are thus calculated with the aid of overlap integrals whose values depend the location of the basis functions; this means that the molecular orbitals and their energies will depend on the actual geometry used in the input, whereas in a simple H€ uckel calculation, the MOs and their energies depend only on the connectivity of the molecule). 4. The overlap matrix S in the EHM is not simply treated as a unit matrix, in effect ignoring it, for the purpose of diagonalizing the Fock matrix. Rather, the overlap integrals are actually evaluated, not only to help calculate the Fock elements, but also to reduce the equation HC ¼ SC« to the standard eigenvalue form HC ¼ C«. This is done in the following way. Suppose the original set of basis functions {fi} could be transformed by some process into an orthonormal set {fi0} (since atom-centered basis functions can’t be orthogonal, as explained in Section 4.3.4, the new set must be delocalized over several centers and is in fact a linear combination of the atom-centered set) such that with a new set of coefficients c0 we have LCAO molecular orbitals with the same energy levels as before, i.e. S0ij ¼

Z

f0i f0j dv ¼ dij

(4.97)

where dij is the Kronecker delta (Eq. 4.57). The result of the process referred to above is Process HC ¼ SC« ! H0 C0 ¼ S0 C0 «

(4.98)

4.4 The Extended H€uckel Method

157

(«, not «0 , as the energy will not depend on manipulation of a given set of basis functions) where the matrices H, C, S and « were defined in Section 4.3.4 (Eqs. 4.55) and H0 and S0 are analogous to H and S with f0 in place of f and C0 is the matrix of coefficients c0 that satisfies the equation with the energy levels e (the elements of «) being the same as in the original equation HC ¼ SC«. Since from Eq. 4.97 S0 ¼ 1, the unit matrix (Section 4.3.3), Eq. 4.98 simplifies to Process HC ¼ SC« ! H0 C0 ¼ C0 «

(4.99)

The Process that effects the transformation is called orthogonalization, since the result is to make the basis functions orthogonal. The favored orthogonalization procedure in computational chemistry, which I will now describe, is L€owdin orthogonalization (after the quantum chemist Per-Olov L€owdin). Define a matrix C0 such that C0 ¼ S1=2 C

i.e: C ¼ S1=2 C0

(4.100)

(By multiplying on the left by S1/2 and noting that S1/2S1/2 ¼ S0 ¼ 1). Substituting Eq. 4.100 into HC ¼ SC« and multiplying on the left by S1/2 we get S1=2 HS1=2 C0 ¼ S1=2 SS1=2 C0 e

(4.101)

S1=2 HS1=2 ¼ H0

(4.102)

Let

and note that S1=2 SS1=2 ¼ S1=2 S1=2 ¼ 1 Then we have from Eqs. 4.101 and 4.102 H0 C0 ¼ 1C0 e i.e. H0 C0 ¼ C0 e

(4.103)

Thus the orthogonalizing process of Eq. 4.99 (or rather one possible orthogonalization process, L€owdin orthogonalization) is the use of an orthogonalizing matrix S1/2 to transform H by pre- and postmultiplication (Eq. 4.102) into H0 . H0 satisfies the standard eigenvalue equation (Eq. 4.103), so

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4 Introduction to Quantum Mechanics in Computational Chemistry 0

H0 ¼ C0 eC 1

(4.104)

In other words, using S1/2 we transform the original Fock matrix H, which is not directly diagonalizable to eigenvector and eigenvalue matrices C and «, into a related matrix H0 which is diagonalizable to eigenvector and eigenvalue matrices C0 and «. The matrix C0 is then transformed to the desired C by multiplying by S1/2 (Eq. 4.100). So without using the drastic S ¼ 1 approximation we can use matrix diagonalization to get the coefficients and energy levels from the Fock matrix. The orthogonalizing matrix S1/2 is calculated from S: the integrals S are calculated and assembled into S, which is then diagonalized: S ¼ PDP1

(4.105)

Now it can be shown that any function of a matrix A can be obtained by taking the same function of its corresponding diagonal alter ego and pre- and postmultiplying by the diagonalizing matrix P and its inverse P1: f ðAÞ ¼ Pf ðDÞP1

(4.106)

and diagonal matrices have the nice property that f(D) is the diagonal matrix whose diagonal element i,,j ¼ f(element i,j of D). So the inverse square root of D is the matrix whose elements are the inverse square roots of the corresponding elements of D. Therefore S1=2 ¼ PD1=2 P1

(4.107)

and to find D1/2 we (or rather the computer) simply take the inverse square root of the diagonal (i.e. the nonzero) elements of D. To summarize: S is diagonalized to give P, P1 and D, D is used to calculate D1/2, then the orthogonalizing matrix S1/2 is calculated (Eq. 4.107) from P, D1/2 and P1. The orthogonalizing matrix is then used to convert H to H0 (Eq. 4.102), which can be diagonalized to give the eigenvalues and the eigenvectors (Section 4.4.2).

4.4.1.3

Review of the EHM Procedure

The EHM procedure for calculating eigenvectors and eigenvalues, i.e. coefficients (or in effect molecular orbitals – the c’s along with the basis functions comprise the MOs) and energy levels, bears several important resemblances to that used in more advanced methods (Chapters 5 and 6) and so is worth reviewing. 1. An input structure (a molecular geometry) must be specified and submitted to calculation. The geometry can be specified in Cartesian coordinates (probably the usual way nowadays) or as bond lengths, angles and dihedrals (internal

4.4 The Extended H€uckel Method

2. 3.

4.

5.

6. 7.

159

coordinates), depending on the program. In practice a virtual molecule would likely be created with an interactive model-building program (usually by clicking together groups and atoms) which would then supply the EHM program with either Cartesian or internal coordinates. The EHM program calculates the overlap integrals S and assembles the overlap matrix S. D E ^ j (Eqs. 4.91 The program calculates the Fock matrix elements Hij ¼ fi jHjf and 4.92) using stored values of ionization energies I, the overlap integrals S, and the proportionality constant K of that particular program. The matrix elements are assembled into the Fock matrix H. The overlap matrix is diagonalized to give P, D and P1 (Eq. 4.105) and D1/2 is then calculated by finding the inverse square roots of the diagonal elements of D. The orthogonalizing matrix S1/2 is then calculated from P, D1/2 and P1 (Eq. 4.107). The Fock matrix H in the atom-centered nonorthogonal basis {f} is transformed into the matrix H0 in the delocalized, linear combination orthogonal basis {f0 } by pre- and postmultiplying H by the orthogonalizing matrix S1/2 (Eq. 4.102). H0 is diagonalized to give C0 , « and C0 1 (Eq. 4.104). We now have the energy levels e (the diagonal elements of the « matrix). C0 must be transformed to give the coefficients c of the original, atom-centered set of basis functions {f} in the MOs (i.e. to convert the elements c0 to c). To get the c’s in the MOs cj ¼ c1jf1 þ c2jf2 þ. . . , we transform C0 to C by premultiplying by S1/2 (Eq. 4.100).

4.4.1.4

Molecular Energy and Geometry Optimization in the Extended H€ uckel Method

Steps 1–7 take an input geometry and calculate its energy levels (the elements of « ) and their MOs or wavefunctions (the c’s; from the c’s, the elements of C, and the basis functions f). Now, clearly any method in which the energy of a molecule depends on its geometry can in principle be used to find minima and transition states (see Chapter 2). This brings us to the matter of how the EHM calculates the energy of a molecule. The energy of a molecule, that is, the energy of a particular nuclear configuration on the potential energy surface, is the sum of the electronic energies and the internuclear repulsions (Eelectronic þ VNN). In comparing the energies of isomers, or of two geometries of the same molecule, one should, strictly, compare Etotal ¼ Eelectronic þ VNN. The electronic energy is the sum of kinetic energy and potential energy (electron–electron repulsion and electron–nucleus attraction) terms. The internuclear repulsion, due to all pairs of interacting nuclei and trivial to calculate, is usually represented by V, a symbol for potential energy. The EHM ignores VNN. Furthermore, the method calculates electronic energy simply as the sum of one-electron energies (Section 4.4.4.2, Weaknesses), ignoring electron–electron repulsion. Hoffmann’s tentative justification

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4 Introduction to Quantum Mechanics in Computational Chemistry

[56a] for ignoring internuclear repulsion and using a simple sum of one-electron energies was that when the relative energies of isomers are calculated, by subtracting two values of Etotal, the electron repulsion and nuclear repulsion terms approximately cancel, i.e. that changes in energy that accompany changes in geometry are due mainly to alterations of the MO energy levels. Actually, it seems that the (quite limited) success of the EHM in predicting molecular geometry is due to the fact that Etotal is approximately proportional to the sum of the occupied MO energies; thus although the EHM energy difference is not equal to the difference in total energies, it is (or tends to be) approximately proportional to this difference [59]. In any case, the real strength of the EHM lies in the ability of this fast and widely applicable method to assist chemical intuition, if provided with a reasonable molecular geometry.

4.4.2

An Illustration of the EHM: the Protonated Helium Molecule

Protonation of a helium atom gives He–H+, the helium hydride cation, the simplest heteronuclear molecule [60]. Conceptually, of course, this can also be formed by the union of a helium dication and a hydride ion, or a helium cation and a hydrogen atom: He: þ Hþ ! He:Hþ or He2þ þ :H ! He:Hþ or Heþ þ H

! He:Hþ

Its lower symmetry makes this molecule better than H2 for illustrating molecular quantum mechanical calculations (most molecules have little or no symmetry). Following the prescription in points 1–7: 1. Input structure ˚ (the H–H bond length is 0.742 A ˚ We choose a plausible bond length: 0.800 A ˚ and the H–X bond length is ca. 1.0 A, where X is a “first-row” element (in quantum chemistry, first-row means Li to F, not H and He). The Cartesian coordinates could be written H1(0,0,0), He2(0,0, 0.800). 2. Overlap integrals and overlap matrix The minimal valence basis set here consists of the hydrogen 1s orbital (f1) and the helium 1s Ð orbital (f2). The needed integrals are S11 ¼ S22 and S12 ¼ S21, where Sij ¼ fi fj dv. The Slater functions for f1 and f2 are [61]

f1 ðH1sÞ ¼

z3H p

!1=2 ezH jrRH j

(4.108)

4.4 The Extended H€uckel Method

161

and

f2 ðHe1sÞ ¼

z3He p

!1=2 ezH jrRHe j

(4.109)

Reasonable values [60] are zH ¼ 1.24 Bohr1 and zHe ¼ 2.0925 Bohr1, if r is in ˚ . The overlap integrals are atomic units, a.u. (see Section 5.2.2); 1 a.u. ¼ 0.5292 A S11 ¼ S22 ¼ 1 (as must be the case if f1 and f2 are R normalized) R and S12 ¼ S21 ¼ 0.435 (for all well-behaved functions f1f2dq ¼ f2f1dq). The overlap matrix is thus
S¼

1 0:435

0:435 1

 (4.110)

3. Fock matrix We need the matrix elements H11 ¼ H22 and H12 ¼ H21, where the integrals ^ j > are not actually calculated from first principles but rather are Hij ¼